Diophantine Equation

Diophantine equation
From Wikipedia, the free encyclopedia

Contents
1

2

3

Amplitude

1

1.1

Definitions of the term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Peak-to-peak amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

Peak amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.3

Semi-amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.4

Root mean square amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.5

Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.6

Pulse amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Formal representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

Waveform and envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.5

Sinusoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.6

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.7

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Arithmetic progression

5

2.1

Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.3

Standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.4

Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.5

Formulas at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.6

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.7

References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.8

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Derivation

Diophantine equation

9

3.1

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.2

Linear Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.2.1

One equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.2.2

Chinese remainder theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3.2.3

System of linear Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Diophantine analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

3.3

i

ii

4

CONTENTS
3.3.1

Typical questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

3.3.2

Typical problem

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.3.3

17th and 18th centuries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.3.4

Hilbert’s tenth problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.3.5

Diophantine geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.3.6

Modern research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.3.7

Infinite Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.4

Exponential Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.5

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.7

Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

3.8

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Equation

15

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

4.1.1

Parameters and unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

4.1.2

Analogous illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.1.3

Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

4.2

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

4.3

Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.3.1

Polynomial equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.3.2

Systems of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

4.4.1

Analytic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

4.4.2

Cartesian equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

4.4.3

Parametric equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

4.5.1

Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

4.5.2

Algebraic and transcendental numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

4.5.3

Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

4.6.1

Ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.6.2

Partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.4

4.5

4.6

5

4.7

Types of equations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.8

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

4.9

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

4.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Geometric progression

27

5.1

Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

5.2

Geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

5.2.1

29

Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS

6

5.2.2

Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

5.2.3

Infinite geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

5.2.4

Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

5.3

Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

5.4

Relationship to geometry and Euclid’s work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

5.5

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

5.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

5.7

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Geometric series

35

6.1

Common ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

6.2

Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

6.2.1

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

6.2.2

Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

6.2.3

Proof of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

6.2.4

Generalized formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

6.3.1

Repeating decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

6.3.2

Archimedes’ quadrature of the parabola . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

6.3.3

Fractal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

6.3.4

Zeno’s paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

6.3.5

Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

6.3.6

Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

6.3.7

Geometric power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

6.4.1

Specific geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

6.5.1

History and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

6.5.2

Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

6.5.3

Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

6.5.4

Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

6.3

6.4
6.5

6.6
7

8

iii

Scale factor

47

7.1

47

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Sequence
8.1

8.2

48

Examples and notation

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49

8.1.1

Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

8.1.2

Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

8.1.3

Specifying a sequence by recursion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Formal definition and basic properties

iv

CONTENTS
8.2.1

Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

8.2.2

Finite and infinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

8.2.3

Increasing and decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

8.2.4

Bounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

8.2.5

Other types of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Limits and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

8.3.1

Definition of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

8.3.2

Applications and important results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

8.3.3

Cauchy sequences

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

8.4

Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

8.5

Use in other fields of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

8.5.1

Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

8.5.2

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

8.5.3

Linear algebra

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

8.5.4

Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

8.5.5

Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

8.5.6

Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

8.5.7

Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

8.6

Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

8.7

Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

8.8

Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

8.9

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

8.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

8.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

8.12 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . .

61

8.12.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

8.12.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

8.12.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

8.3

Chapter 1

Amplitude
This article is about amplitude in classical physics. For quantum-mechanical amplitude, see probability amplitude.
For the video game, see Amplitude (video game).
The amplitude of a periodic variable is a measure of its change over a single period (such as time or spatial period).
There are various definitions of amplitude (see below), which are all functions of the magnitude of the difference
between the variable’s extreme values. In older texts the phase is sometimes called the amplitude.[1]

1.1 Definitions of the term
1.1.1

Peak-to-peak amplitude

“Peak to peak” redirects here. For the school, see Peak to Peak Charter School.
Peak-to-peak amplitude is the change between peak (highest amplitude value) and trough (lowest amplitude value,
which can be negative). With appropriate circuitry, peak-to-peak amplitudes of electric oscillations can be measured
by meters or by viewing the waveform on an oscilloscope. Peak-to-peak is a straightforward measurement on an
oscilloscope, the peaks of the waveform being easily identified and measured against the graticule. This remains a
common way of specifying amplitude, but sometimes other measures of amplitude are more appropriate.

1.1.2

Peak amplitude

In audio system measurements, telecommunications and other areas where the measurand is a signal that swings above
and below a zero value but is not sinusoidal, peak amplitude is often used. This is the maximum absolute value of
the signal.

1.1.3

Semi-amplitude

Semi-amplitude means half the peak-to-peak amplitude.[2] It is the most widely used measure of orbital amplitude in
astronomy and the measurement of small semi-amplitudes of nearby stars is important in the search for exoplanets.[3]
Some scientists[4] use “amplitude” or “peak amplitude” to mean semi-amplitude, that is, half the peak-to-peak amplitude.

1.1.4

Root mean square amplitude

Root mean square (RMS) amplitude is used especially in electrical engineering: the RMS is defined as the square
root of the mean over time of the square of the vertical distance of the graph from the rest state.[5]
1

2

CHAPTER 1. AMPLITUDE

u
+Û
1

3
0

-Û

2
t

4

A sinusoidal curve
1 = Peak amplitude ( Uˆ ),
2 = Peak-to-peak amplitude ( 2Uˆ ),
√
3 = Root mean square amplitude ( Uˆ / 2 ),
4 = Wave period (not an amplitude)

For complicated waveforms, especially non-repeating signals like noise, the RMS amplitude is usually used because
it is both unambiguous and has physical significance. For example, the average power transmitted by an acoustic
or electromagnetic wave or by an electrical signal is proportional to the square of the RMS amplitude (and not, in
general, to the square of the peak amplitude).[6]
For alternating current electric power, the universal practice is to specify RMS values of a sinusoidal waveform. One
property of root mean square voltages and currents is that they produce the same heating effect as direct current in a
given resistance.
The peak-to-peak voltage of a sine wave is about 2.8 times the RMS value. The peak-to-peak value is used, for
example, when choosing rectifiers for power supplies, or when estimating the maximum voltage that insulation must
withstand. Some common voltmeters are calibrated for RMS amplitude, but respond to the average value of a rectified
waveform. Many digital voltmeters and all moving coil meters are in this category. The RMS calibration is only
correct for a sine wave input since the ratio between peak, average and RMS values is dependent on waveform. If the
wave shape being measured is greatly different from a sine wave, the relationship between RMS and average value
changes. True RMS-responding meters were used in radio frequency measurements, where instruments measured the
heating effect in a resistor to measure current. The advent of microprocessor controlled meters capable of calculating
RMS by sampling the waveform has made true RMS measurement commonplace.

1.1.5

Ambiguity

In general, the use of peak amplitude is simple and unambiguous only for symmetric periodic waves, like a sine
wave, a square wave, or a triangular wave. For an asymmetric wave (periodic pulses in one direction, for example),
the peak amplitude becomes ambiguous. This is because the value is different depending on whether the maximum
positive signal is measured relative to the mean, the maximum negative signal is measured relative to the mean, or

1.2. FORMAL REPRESENTATION

3

the maximum positive signal is measured relative to the maximum negative signal (the peak-to-peak amplitude) and
then divided by two. In electrical engineering, the usual solution to this ambiguity is to measure the amplitude from
a defined reference potential (such as ground or 0 V). Strictly speaking, this is no longer amplitude since there is the
possibility that a constant (DC component) is included in the measurement.

1.1.6

Pulse amplitude

In telecommunication, pulse amplitude is the magnitude of a pulse parameter, such as the voltage level, current level,
field intensity, or power level.
Pulse amplitude is measured with respect to a specified reference and therefore should be modified by qualifiers, such
as “average”, “instantaneous”, “peak”, or “root-mean-square”.
Pulse amplitude also applies to the amplitude of frequency- and phase-modulated waveform envelopes.[7]

1.2 Formal representation
In this simple wave equation

x = A sin(ω(t − K)) + b ,
A is the peak amplitude of the wave,
x is the oscillating variable,
ω is angular frequency,
t is time,
K and b are arbitrary constants representing time and displacement offsets respectively.

1.3 Units
The units of the amplitude depend on the type of wave, but are always in the same units as the oscillating variable. A
more general representation of the wave equation is more complex, but the role of amplitude remains analogous to
this simple case.
For waves on a string, or in medium such as water, the amplitude is a displacement.
The amplitude of sound waves and audio signals (which relates to the volume) conventionally refers to the amplitude
of the air pressure in the wave, but sometimes the amplitude of the displacement (movements of the air or the
diaphragm of a speaker) is described. The logarithm of the amplitude squared is usually quoted in dB, so a null
amplitude corresponds to −∞ dB. Loudness is related to amplitude and intensity and is one of most salient qualities
of a sound, although in general sounds can be recognized independently of amplitude. The square of the amplitude
is proportional to the intensity of the wave.
For electromagnetic radiation, the amplitude of a photon corresponds to the changes in the electric field of the wave.
However radio signals may be carried by electromagnetic radiation; the intensity of the radiation (amplitude modulation) or the frequency of the radiation (frequency modulation) is oscillated and then the individual oscillations are
varied (modulated) to produce the signal.

1.4 Waveform and envelope
Main article: Envelope (waves)
The amplitude as defined above is a constant and the wave is said to be continuous. If this condition does not hold,
amplitude-like variations with time and/or position may be quantified in terms of the envelope of the wave.

4

CHAPTER 1. AMPLITUDE

1.5 Sinusoids
If the waveform is a pure sine wave, the relationships between peak-to-peak, peak, mean, and RMS amplitudes are
fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform
which may or may not be periodic or continuous.
For a sine wave, the relationship between RMS and peak-to-peak amplitude is:
√
Peak-to-peak = 2 2 × RMS ≈ 2.8 × RMS
See Root mean square#RMS of common waveforms for more.
For other waveforms the relationships are not (necessarily) arithmetically the same as they are for sine waves.

1.6 See also
• Complex amplitude
• Waves and their properties:
• Frequency
• Wavelength
• Crest factor
• Amplitude modulation

1.7 Notes
[1] Knopp, Konrad; Bagemihl, Frederick (1996). Theory of Functions Parts I and II. Dover Publications. p. 3. ISBN 978-0486-69219-7.
[2] Tatum, J. B. Physics – Celestial Mechanics. Paragraph 18.2.12. 2007. Retrieved 2008-08-22.
[3] Goldvais, Uriel A. Exoplanets, pp. 2–3. Retrieved 2008-08-22.
[4] Regents of the University of California. Universe of Light: What is the Amplitude of a Wave? 1996. Retrieved 2008-08-22.
[5] Department of Communicative Disorders University of Wisconsin–Madison. RMS Amplitude. Retrieved 2008-08-22.
[6] Ward, Electrical Engineering Science, pp. 141–142, McGraw-Hill, 1971.
[7] This article incorporates public domain material from the General Services Administration document “Federal Standard
1037C”.

Chapter 2

Arithmetic progression
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the
difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15 … is an arithmetic
progression with common difference of 2.
If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the
nth term of the sequence ( an ) is given by:
an = a1 + (n − 1)d,
and in general

an = am + (n − m)d.
A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an
arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.
The behavior of the arithmetic progression depends on the common difference d. If the common difference is:
• Positive, the members (terms) will grow towards positive infinity.
• Negative, the members (terms) will grow towards negative infinity.

2.1 Sum
This section is about Finite arithmetic series. For Infinite arithmetic series, see Infinite arithmetic series.
Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the
resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers (2 + 14 = 16). Thus
16 × 5 = 80 is twice the sum.
The sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the
sum:

2 + 5 + 8 + 11 + 14
This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the
first and last number in the progression (here 2 + 14 = 16), and dividing by 2:
n(a1 + an )
2
5

6

CHAPTER 2. ARITHMETIC PROGRESSION

In the case above, this gives the equation:

2 + 5 + 8 + 11 + 14 =

5(2 + 14)
5 × 16
=
= 40.
2
2

This formula works for any real numbers a1 and an . For example:
(
)
(
) (
)
3 − 32 + 12
3
1
1
3
−
+ −
+ =
=− .
2
2
2
2
2

2.1.1

Derivation

To derive the above formula, begin by expressing the arithmetic series in two different ways:

Sn = a1 + (a1 + d) + (a1 + 2d) + · · · + (a1 + (n − 2)d) + (a1 + (n − 1)d)
Sn = (an − (n − 1)d) + (an − (n − 2)d) + · · · + (an − 2d) + (an − d) + an .
Adding both sides of the two equations, all terms involving d cancel:

2Sn = n(a1 + an ).
Dividing both sides by 2 produces a common form of the equation:

Sn =

n
(a1 + an ).
2

An alternate form results from re-inserting the substitution: an = a1 + (n − 1)d :

Sn =

n
[2a1 + (n − 1)d].
2

Furthermore the mean value of the series can be calculated via: Sn /n :

n=

a1 + an
.
2

In 499 AD Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics and
Indian astronomy, gave this method in the Aryabhatiya (section 2.18).

2.2 Product
The product of the members of a finite arithmetic progression with an initial element a1 , common differences d, and
n elements in total is determined in a closed expression

a1 a2 · · · an = d

( a )n
a1
a1
Γ (a1 /d + n)
a1 a1
1
d( + 1)d( + 2) · · · d( + n − 1) = dn
,
= dn
d
d
d
d
d
Γ (a1 /d)

where xn denotes the rising factorial and Γ denotes the Gamma function. (Note however that the formula is not valid
when a1 /d is a negative integer or zero.)
This is a generalization from the fact that the product of the progression 1 × 2 × · · · × n is given by the factorial n!
and that the product

2.3. STANDARD DEVIATION

7

m × (m + 1) × (m + 2) × · · · × (n − 2) × (n − 1) × n
for positive integers m and n is given by
n!
.
(m − 1)!
Taking the example from above, the product of the terms of the arithmetic progression given by an = 3 + (n−1)(5)
up to the 50th term is

P50 = 550 ·

Γ (3/5 + 50)
≈ 3.78438 × 1098 .
Γ (3/5)

2.3 Standard deviation
The standard deviation of any arithmetic progression can be calculated via:
√

(n − 1)(n + 1)
12
where n is the number of terms in the progression, and d is the common difference between terms
σ = |d|

2.4 Intersections
The intersection of any two doubly-infinite arithmetic progressions is either empty or another arithmetic progression,
which can be found using the Chinese remainder theorem. If each two progressions in a family of doubly-infinite
arithmetic progressions have a non-empty intersection, then there exists a number common to all of them; that is,
infinite arithmetic progressions form a Helly family.[1] However, the intersection of infinitely many infinite arithmetic
progressions might be a single number rather than itself being an infinite progression.

2.5 Formulas at a Glance
If

a1
an
d
n
Sn
n
then
an = a1 + (n − 1)d,
an = am + (n − m)d.
n
Sn = [2a1 + (n − 1)d].
2
n(a1 + an )
Sn =
2

8

CHAPTER 2. ARITHMETIC PROGRESSION
5. n = Sn /n

n=

a1 + an
.
2

2.6 See also
• Arithmetico-geometric sequence
• Generalized arithmetic progression - is a set of integers constructed as an arithmetic progression is, but allowing
several possible differences.
• Harmonic progression
• Heronian triangles with sides in arithmetic progression
• Problems involving arithmetic progressions
• Utonality

2.7 References
[1] Duchet, Pierre (1995), “Hypergraphs”, in Graham, R. L.; Grötschel, M.; Lovász, L., Handbook of combinatorics, Vol. 1,
2, Amsterdam: Elsevier, pp. 381–432, MR 1373663. See in particular Section 2.5, “Helly Property”, pp. 393–394.

• Sigler, Laurence E. (trans.) (2002). Fibonacci’s Liber Abaci. Springer-Verlag. pp. 259–260. ISBN 0-38795419-8.

2.8 External links
• Hazewinkel, Michiel, ed. (2001), “Arithmetic series”, Encyclopedia of Mathematics, Springer, ISBN 978-155608-010-4
• Weisstein, Eric W., “Arithmetic progression”, MathWorld.
• Weisstein, Eric W., “Arithmetic series”, MathWorld.

Chapter 3

Diophantine equation

Finding all right triangles with integer side-lengths is equivalent to solving the Diophantine equation a2 + b2 = c2 .

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only
the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer
values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An
exponential Diophantine equation is one in which exponents on terms can be unknowns.
Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly
for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object,
and ask about the lattice points on it.
9

10

CHAPTER 3. DIOPHANTINE EQUATION

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who
made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The
mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.
While individual equations present a kind of puzzle and have been considered throughout history, the formulation of
general theories of Diophantine equations (beyond the theory of quadratic forms) was an achievement of the twentieth
century.

3.1 Examples
In the following Diophantine equations, w, x, y, and z are the unknowns and the other letters are given constants:

3.2 Linear Diophantine equations
3.2.1

One equation

The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers. The
solutions are completely described by the following theorem: This Diophantine equation has a solution (where x and
y are integers) if and only if c is a multiple of the greatest common divisor of a and b. Moreover, if (x, y) is a solution,
then the other solutions have the form (x + kv, y - ku), where k is an arbitrary integer, and u and v are the quotients of
a and b (respectively) by the greatest common divisor of a and b.
Proof: If d is this greatest common divisor, Bézout’s identity asserts the existence of integers e and f such that ae +
bf = d. If c is a multiple of d, then c = dh for some integer h, and (eh, fh) is a solution. On the other hand, for every
integers x and y, the greatest common divisor d of a and b divides ax + by. Thus, if the equation has a solution, then
c must be a multiple of d. If a = ud and b = vd, then for every solution (x, y), we have

a(x + kv) + b(y − ku) = ax + by + k(av − bu) = ax + by + k(udv − vdu) = ax + by
showing that (x + kv, y - ku) is another solution. Finally, given two solutions such that ax1 + by1 = ax2 + by2 = c, one
deduces that u (x2 - x1 ) + v (y2 - y1 ) = 0. As u and v are coprime, Euclid’s lemma shows that there exists an integer
k such that x2 - x1 = kv and y2 - y1 = -ku. Therefore x2 = x1 + kv and y2 = y1 - ku, which completes the proof.

3.2.2

Chinese remainder theorem

The Chinese remainder theorem describes an important class of linear Diophantine systems of equations: let n1 , ...,
nk be k pairwise coprime integers greater than one, a1 , ..., ak be k arbitrary integers, and N be the product n1 ··· nk.
The Chinese remainder theorem asserts that the following linear Diophantine system has exactly one solution (x, x1 ,
..., xk) such that 0 ≤ x < N, and that the other solutions are obtained by adding to x a multiple of N:

x = a1 + n1 x1
..
.
x = ak + nk xk

3.2.3

System of linear Diophantine equations

More generally, every system of linear Diophantine equations may be solved by computing the Smith normal form
of its matrix, in a way that is similar to the use of the reduced row echelon form to solve a system of linear equations
over a field. Using matrix notation every system of linear Diophantine equations may be written

3.3. DIOPHANTINE ANALYSIS

11

A X = C,
where A is an m×n matrix of integers, X is an n×1 column matrix of unknowns and C is an m×1 column matrix of
integers.
The computation of the Smith normal form of A provides two unimodular matrices (that is matrices that are invertible
over the integers, which have ±1 as determinant) U and V of respective dimensions m×m and n×n, such that the matrix

B = [bi,j ] = U AV
is such that bi,i is not zero for i not greater than some integer k, and all the other entries are zero. The system to be
solved may thus be rewritten as

B (V −1 X) = U C.
Calling yi the entries of V −1 X and di those of D = U C, this leads to the system
bi,i yi = di for 1 ≤ i ≤ k,
0 yi = di for k < i ≤ n.
This system is equivalent to the given one in the following sense: A column matrix of integers x is a solution of the
given system if and only if x = V y for some column matrix of integers y such that By = D.
It follows that the system has a solution if and only if bi,i divides di for i ≤ k and di = 0 for i > k. If this condition is
fulfilled, the solutions of the given system are


d1
b1,1

 .
 .
 .
 dk

V  bk,k
 hk+1

 ..
 .
hn







,





where hk₊₁, ..., hn are arbitrary integers.
Hermite normal form may also be used for solving systems of linear Diophantine equations. However, Hermite
normal form does not provide directly the solutions; for getting the solutions from the Hermite normal form, one
has to solve successively several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form
“is somewhat more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to
diagonal form, we only need to make it triangular, which is called the Hermite normal form. The Hermite normal
form is substantially easier to compute than the Smith normal form.”[5]
Integer linear programming amounts to find some integer solutions (optimal in some sense) of linear systems that
include also inequations. Thus systems of linear Diophantine equations are basic in this context, and textbooks on
integer programming usually have a treatment of systems of linear Diophantine equations.[6]

3.3 Diophantine analysis
3.3.1

Typical questions

The questions asked in Diophantine analysis include:
1. Are there any solutions?

12

CHAPTER 3. DIOPHANTINE EQUATION
2. Are there any solutions beyond some that are easily found by inspection?
3. Are there finitely or infinitely many solutions?
4. Can all solutions be found in theory?
5. Can one in practice compute a full list of solutions?

These traditional problems often lay unsolved for centuries, and mathematicians gradually came to understand their
depth (in some cases), rather than treat them as puzzles.

3.3.2

Typical problem

The given information is that a father’s age is 1 less than twice that of his son, and that the digits AB making up the
father’s age are reversed in the son’s age (i.e. BA). This leads to the equation 10A + B = 2 (10B + A) - 1, thus 19B 8A = 1. Inspection gives the result A = 7, B = 3, and thus AB = 73 years and BA = 37 years. One may easily show
that there is not any other solution with A and B positive integers less than 10.

3.3.3

17th and 18th centuries

In 1637, Pierre de Fermat scribbled on the margin of his copy of Arithmetica: “It is impossible to separate a cube into
two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like
powers.” Stated in more modern language, “The equation an + bn = cn has no solutions for any n higher than 2.” And
then he wrote, intriguingly: “I have discovered a truly marvelous proof of this proposition, which this margin is too
narrow to contain.” Such a proof eluded mathematicians for centuries, however, and as such his statement became
famous as Fermat’s Last Theorem. It wasn't until 1995 that it was proven by the British mathematician Andrew Wiles.
In 1657, Fermat attempted to solve the Diophantine equation 61x2 + 1 = y2 (solved by Brahmagupta over 1000 years
earlier). The equation was eventually solved by Euler in the early 18th century, who also solved a number of other
Diophantine equations.The smallest solution of this equation in positive integers is x = 226153980, y = 1766319049
(see Chakravala method).

3.3.4

Hilbert’s tenth problem

In 1900, in recognition of their depth, David Hilbert proposed the solvability of all Diophantine problems as the tenth
of his celebrated problems. In 1970, a novel result in mathematical logic known as Matiyasevich’s theorem settled
the problem negatively: in general Diophantine problems are unsolvable.

3.3.5

Diophantine geometry

Diophantine geometry, which is the application of techniques from algebraic geometry in this field, has continued
to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations that also have a
geometric meaning. The central idea of Diophantine geometry is that of a rational point, namely a solution to a
polynomial equation or a system of polynomial equations, which is a vector in a prescribed field K, when K is not
algebraically closed.

3.3.6

Modern research

One of the few general approaches is through the Hasse principle. Infinite descent is the traditional method, and has
been pushed a long way.
The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as
equivalently described as recursively enumerable. In other words, the general problem of Diophantine analysis is
blessed or cursed with universality, and in any case is not something that will be solved except by re-expressing it in
other terms.

3.4. EXPONENTIAL DIOPHANTINE EQUATIONS

13

The field of Diophantine approximation deals with the cases of Diophantine inequalities. Here variables are still
supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper
and lower bounds.
The most celebrated single question in the field, the conjecture known as Fermat’s Last Theorem, was solved by
Andrew Wiles[7] but using tools from algebraic geometry developed during the last century rather than within number
theory where the conjecture was originally formulated. Other major results, such as Faltings’ theorem, have disposed
of old conjectures.

3.3.7

Infinite Diophantine equations

An example of an infinite diophantine equation is:

N = A2 + 2B 2 + 3C 2 + 4D2 + 5E 2 + ...,
which can be expressed as “How many ways can a given integer N be written as the sum of a square plus twice a
square plus thrice a square and so on?" The number of ways this can be done for each N forms an integer sequence.
Infinite Diophantine equations are related to theta functions and infinite dimensional lattices. This equation always
has a solution for any positive N. Compare this to:

N = A2 + 4B 2 + 9C 2 + 16D2 + 25E 2 + ...,
which does not always have a solution for positive N.

3.4 Exponential Diophantine equations
If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential
Diophantine equation. Examples include the Ramanujan–Nagell equation, 2n − 7 = x2 , and the equation of the
Fermat-Catalan conjecture and Beal’s conjecture, am + bn = ck with inequality restrictions on the exponents. A
general theory for such equations is not available; particular cases such as Catalan’s conjecture have been tackled.
However, the majority are solved via ad hoc methods such as Størmer’s theorem or even trial and error.

3.5 Notes
[1] “Quotations by Hardy”. Gap.dcs.st-and.ac.uk. Retrieved 20 November 2012.
[2] Everest, G.; Ward, Thomas (2006), An Introduction to Number Theory, Graduate Texts in Mathematics 232, Springer, p.
117, ISBN 9781846280443.
[3] Wiles, Andrew (1995). “Modular elliptic curves and Fermat’s Last Theorem” (PDF). Annals of Mathematics (Annals of
Mathematics) 141 (3): 443–551. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255.
[4] Noam Elkies (1988). “On A4 + B4 + C 4 = D4 ". Mathematics of Computation 51 (184): 825–835. doi:10.2307/2008781.
JSTOR 2008781. MR 0930224.
[5] Richard Zippel (1993). Effective Polynomial Computation. Springer Science & Business Media. p. 50. ISBN 978-0-79239375-7.
[6] Alexander Bockmayr, Volker Weispfenning (2001). “Solving Numerical Constraints”. In John Alan Robinson and Andrei
Voronkov. Handbook of Automated Reasoning Volume I. Elsevier and MIT Press. p. 779. ISBN 0-444-82949-0 (Elsevier)
ISBN 0-262-18221-1 (MIT Press).
[7] Solving Fermat: Andrew Wiles

14

CHAPTER 3. DIOPHANTINE EQUATION

3.6 References
• Mordell, L. J. (1969). Diophantine equations. Pure and Applied Mathematics 30. Academic Press. ISBN
0-12-506250-8. Zbl 0188.34503.
• Schmidt, Wolfgang M. (1991). Diophantine approximations and Diophantine equations. Lecture Notes in
Mathematics 1467. Berlin: Springer-Verlag. ISBN 3-540-54058-X. Zbl 0754.11020.
• Shorey, T. N.; Tijdeman, R. (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics
87. Cambridge University Press. ISBN 0-521-26826-5. Zbl 0606.10011.
• Smart, Nigel P. (1998). The algorithmic resolution of Diophantine equations. London Mathematical Society
Student Texts 41. Cambridge University Press. ISBN 0-521-64156-X. Zbl 0907.11001.
• Stillwell, John (2004). Mathematics and its History (Second Edition ed.). Springer Science + Business Media
Inc. ISBN 0-387-95336-1.

3.7 Further reading
• Dickson, Leonard Eugene (2005) [1920]. History of the Theory of Numbers. Volume II: Diophantine analysis.
Mineola, NY: Dover Publications. ISBN 978-0-486-44233-4. MR 0245500. Zbl 1214.11002.

3.8 External links
• Diophantine Equation. From MathWorld at Wolfram Research.
• Diophantine Equation. From PlanetMath.
• Hazewinkel, Michiel, ed. (2001), “Diophantine equations”, Encyclopedia of Mathematics, Springer, ISBN
978-1-55608-010-4
• Dario Alpern’s Online Calculator. Retrieved 18 March 2009

Chapter 4

Equation
For other uses, see Equation (disambiguation).
In mathematics, an equation is an equality containing one or more variables. Solving the equation consists of

The first use of an equals sign, equivalent to 14x + 15 = 71 in modern notation. From The Whetstone of Witte by Robert Recorde
(1557).

determining which values of the variables make the equality true. In this situation, variables are also known as
unknowns and the values which satisfy the equality are known as solutions. An equation differs from an identity in
that an equation is not necessarily true for all possible values of the variable.[1][2]
There are many types of equations, and they are found in all areas of mathematics; the techniques used to examine
them differ according to their type.
Algebra studies two main families of equations: polynomial equations and, among them, linear equations. Polynomial
equations have the form P(X) = 0, where P is a polynomial. Linear equations have the form a(x) + b = 0, where a is
a linear function and b is a vector. To solve them, one uses algorithmic or geometric techniques, coming from linear
algebra or mathematical analysis. Changing the domain of a function can change the problem considerably. Algebra
also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different
and come from number theory. These equations are difficult in general; one often searches just to find the existence
or absence of a solution, and, if they exist, to count the number of solutions.
Geometry uses equations to describe geometric figures. The objective is now different, as equations are used to
describe geometric properties. In this context, there are two large families of equations, Cartesian equations and
parametric equations.
Differential equations are equations involving one or more functions and their derivatives. They are solved by finding
an expression for the function that does not involve derivatives. Differential equations are used to model real-life
processes in areas such as physics, chemistry, biology, and economics.
The "=" symbol was invented by Robert Recorde (1510–1558), who considered that nothing could be more equal
than parallel straight lines with the same length.

4.1 Introduction
4.1.1

Parameters and unknowns

See also: Expression (mathematics)

15

16

CHAPTER 4. EQUATION

A strange attractor which arises when solving a certain differential equation.

Equations often contain terms other than the unknowns. These other terms, which are assumed to be known, are
usually called constants, coefficients or parameters. Usually, the unknowns are denoted by letters at the end of the
alphabet, x, y, z, w, …, while coefficients are denoted by letters at the beginning, a, b, c, d, … . For example, the
general quadratic equation is usually written ax2 + bx + c = 0. The process of finding the solutions, or in case of
parameters, expressing the unknowns in terms of the parameters is called solving the equation. Such expressions of
the solutions in terms of the parameters are also called solutions.
A system of equations is a set of simultaneous equations, usually in several unknowns, for which the common solutions
are sought. Thus a solution to the system is a set of values for each of the unknowns, which together form a solution
to each equation in the system. For example, the system

3x + 5y = 2
5x + 8y = 3
has the unique solution x = −1, y = 1.

4.1.2

Analogous illustration

A weighing scale, balance, or seesaw is often presented as an analogy to an equation.
Each side of the balance corresponds to one side of the equation. Different quantities can be placed on each side: if
the weights on the two sides are equal the scale balances, corresponding to an equality represented by an equation; if
not, then the lack of balance corresponds to an inequality represented by an inequation.
In the illustration, x, y and z are all different quantities (in this case real numbers) represented as circular weights,
and each of x, y, and z has a different weight. Addition corresponds to adding weight, while subtraction corresponds

4.1. INTRODUCTION

17

z
z

+

+ +
x

y

=

y y

x

3y

−
1

1

2z

Illustration of a simple equation; x, y, z are real numbers, analogous to weights.

to removing weight from what is already there. When equality holds, the total weight on each side is the same.

4.1.3

Identities

Main articles: Identity (mathematics) and List of trigonometric identities
An identity is a statement resembling an equation which is true for all possible values of the variable(s) it contains.
Many identities are known in algebra and calculus. In the process of solving an equation, it is often useful to combine
it with an identity to produce an equation which is more easily soluble.
In algebra, a simple identity is the difference of two squares:

x2 − y 2 = (x + y)(x − y)
which is true for all x and y.

18

CHAPTER 4. EQUATION

Trigonometry is an area where many identities exist, and are useful in manipulating or solving trigonometric equations,
two of many including the sine and cosine functions are:

sin2 (θ) + cos2 (θ) = 1
and

sin(2θ) = 2 sin(θ) cos(θ)
which are both true for all values of θ.
For example, to solve the equation:

3 sin(θ) cos(θ) = 1 ,
where θ is known to be between 0 and 45 degrees, using the identity for the product gives
3
sin(2θ) = 1 ,
2
yielding the solution

θ=

1
arcsin
2

( )
2
≈ 20.9◦ .
3

Since the sine function is a periodic function, there are infinitely many solutions if there are no restrictions on θ. In
this example, the fact that θ is between 0 and 45 degrees implies there is only one solution.

4.2 Properties
Two equations or two systems of equations are equivalent if they have the same set of solutions. The following
operations transform an equation or a system into an equivalent one:
• Adding or subtracting the same quantity to both sides of an equation. This shows that every equation is equivalent to an equation in which the right-hand side is zero.
• Multiplying or dividing both sides of an equation by a non-zero constant.
• Applying an identity to transform one side of the equation. For example, expanding a product or factoring a
sum.
• For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by
the same quantity.
If some function is applied to both sides of an equation, the resulting equation has the solutions of the initial equation
among its solutions, but may have further solutions called extraneous solutions. For example, the equation x = 1 has
the solution x = 1. Raising both sides to the exponent of 2 (which means applying the function f (s) = s2 to both
sides of the equation) changes the equation to x2 = 1 , which not only has the previous solution but also introduces
the extraneous solution, x = −1. Moreover, If the function is not defined at some values (such as 1/x, which is not
defined for x = 0), solutions existing at those values may be lost. Thus, caution must be exercised when applying such
a transformation to an equation.
The above transformations are the basis of most elementary methods for equation solving as well as some less elementary ones, like Gaussian elimination.
For more details on this topic, see Equation solving.

4.3. ALGEBRA

19

4.3 Algebra
4.3.1

Polynomial equations

Main article: Polynomial equation
An algebraic equation or polynomial equation is an equation of the form

P =0
P =Q
where P and Q are polynomials with coefficients in some field, often the field of the rational numbers. An algebraic
equation is univariate if it involves only one variable. On the other hand, a polynomial equation may involve several
variables, in which case it is called multivariate and the term polynomial equation is usually preferred to algebraic
equation.
For example,

x5 − 3x + 1 = 0
is an algebraic equation with integer coefficients and

y4 +

xy
x3
1
=
− xy 2 + y 2 −
2
3
7

is a multivariate polynomial equation over the rationals.
Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression with
a finite number of operations involving just those coefficients (that is, can be solved algebraically). This can be done
for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some
equations but not for all. A large amount of research has been devoted to compute efficiently accurate approximations
of the real or complex solutions of an univariate algebraic equation (see Root-finding algorithm) and of the common
solutions of several multivariate polynomial equations (see System of polynomial equations).

4.3.2

Systems of linear equations

A system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.[3]
For example,
3x + 2y − z = 1
2x − 2y + 4z = −2
−x + 12 y − z =

0

is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of numbers
to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by

x = 1
y = −2
z = −2
since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually.

20

CHAPTER 4. EQUATION

The Nine Chapters on the Mathematical Art is an anonymous Chinese book proposing a method of resolution for linear equations.

In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which
is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important
part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and
economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a
helpful technique when making a mathematical model or computer simulation of a relatively complex system.

4.4. GEOMETRY

21

4.4 Geometry
4.4.1

Analytic geometry

A conic section is the intersection of a plane and a cone of revolution.

In Euclidean geometry, it is possible to associate a set of coordinates to each point in space, for example by an
orthogonal grid. This method allows one to characterize geometric figures by equations. A plane in three-dimensional
space can be expressed as the solution set of an equation of the form ax + by + cz + d = 0 , where a, b, c and d are
real numbers and x, y, z are the unknowns which correspond to the coordinates of a point in the system given by the
orthogonal grid. The values a, b, c are the coordinates of a vector perpendicular to the plane defined by the equation.
A line is expressed as the intersection of two planes, that is as the solution set of a single linear equation with values
in R⊭ or as the solution set of two linear equations with values in R .
A conic section is the intersection of a cone with equation x2 + y 2 = z 2 and a plane. In other words, in space,
all conics are defined as the solution set of an equation of a plane and of the equation of a plane just given. This
formalism allows one to determine the positions and the properties of the focuses of a conic.
The use of equations allows one to call on a large area of mathematics to solve geometric questions. The Cartesian
coordinate system transforms a geometric problem into an analysis problem, once the figures are transformed into
equations; thus the name analytic geometry. This point of view, outlined by Descartes, enriches and modifies the
type of geometry conceived of by the ancient Greek mathematicians.
Currently, analytic geometry designates an active branch of mathematics. Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as functional analysis and linear algebra.

4.4.2

Cartesian equations

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of
numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines,
measured in the same unit of length.
One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection
onto three mutually perpendicular lines).

22

CHAPTER 4. EQUATION

y
3
2

2

2

x +y = 4

1
-3

-2

-1

1
-1

2

3

x

-2
-3

Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a)2 + (y
− b)2 = r2 where a and b are the coordinates of the center (a, b) and r is the radius.

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the
Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic
equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane may
be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.

4.4.3

Parametric equations

Main article: Parametric equation
A parametric equation for a curve expresses the coordinates of the points of the curve as functions of a variable,
called a parameter.[4][5] For example,

x = cos t
y = sin t

4.5. NUMBER THEORY

23

are parametric equations for the unit circle, where t is the parameter. Together, these equations are called a parametric representation of the curve.
The notion of parametric equation has been generalized to surfaces, manifolds and algebraic varieties of higher
dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number
of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the
dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).

4.5 Number theory
4.5.1

Diophantine equations

Main article: Diophantine equation
A Diophantine equation is a polynomial equation in two or more unknowns such that only the integer solutions
are searched or studied (an integer solution is a solution such that all the unknowns take integer values). A linear
Diophantine equation is an equation between two sums of monomials of degree zero or one. An exponential
Diophantine equation is one in which exponents on terms can be unknowns.
Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly
for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object,
and ask about the lattice points on it.
The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who
made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The
mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

4.5.2

Algebraic and transcendental numbers

Main articles: Algebraic number and Transcendental number
An algebraic number is a number that is a root of a non-zero polynomial equation in one variable with rational
coefficients (or equivalently — by clearing denominators — with integer coefficients). Numbers such as π that are
not algebraic are said to be transcendental. Almost all real and complex numbers are transcendental.

4.5.3

Algebraic geometry

Main article: Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of polynomial equations. Modern algebraic
geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language
and the problems of geometry.
The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations
of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are:
plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves
and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its
coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like
the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of
the curve and relations between the curves given by different equations.

4.6 Differential equations
Main article: Differential equation

24

CHAPTER 4. EQUATION

A differential equation is a mathematical equation that relates some function with its derivatives. In applications,
the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation
defines a relationship between the two. Because such relations are extremely common, differential equations play a
prominent role in many disciplines including engineering, physics, economics, and biology.
In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with
their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by
explicit formulas; however, some properties of solutions of a given differential equation may be determined without
finding their exact form.
If a self-contained formula for the solution is not available, the solution may be numerically approximated using
computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of
accuracy.

4.6.1

Ordinary differential equations

Main article: Ordinary differential equation
An ordinary differential equation or ODE is an equation containing a function of one independent variable and its
derivatives. The term "ordinary" is used in contrast with the term partial differential equation which may be with
respect to more than one independent variable.
Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined
and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are
nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed
form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods,
applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often
sufficing in the absence of exact, analytic solutions.

4.6.2

Partial differential equations

Main article: Partial differential equation
A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and
their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single
variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are
either solved by hand, or used to create a relevant computer model.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid
flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly
in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial
differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial
differential equations.

4.7 Types of equations
Equations can be classified according to the types of operations and quantities involved. Important types include:
• An algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also
system of polynomial equations). These are further classified by degree:
• linear equation for degree one
• quadratic equation for degree two
• cubic equation for degree three
• quartic equation for degree four

4.8. SEE ALSO

25

• quintic equation for degree five
• sextic equation for degree six
• septic equation for degree seven
• A Diophantine equation is an equation where the unknowns are required to be integers
• A transcendental equation is an equation involving a transcendental function of its unknowns
• A parametric equation is an equation for which the solutions are sought as functions of some other variables,
called parameters appearing in the equations
• A functional equation is an equation in which the unknowns are functions rather than simple quantities
• A differential equation is a functional equation involving derivatives of the unknown functions
• An integral equation is a functional equation involving the antiderivatives of the unknown functions
• An integro-differential equation is a functional equation involving both the derivatives and the antiderivatives
of the unknown functions
• A difference equation is an equation where the unknown is a function f which occurs in the equation through
f(x), f(x−1), …, f(x−k), for some whole integer k called the order of the equation. If x is restricted to be an
integer, a difference equation is the same as a recurrence relation

4.8 See also
• Equation (poem)
• Expression
• Five Equations That Changed the World: The Power and Poetry of Mathematics (book)
• Formula
• Formula editor
• Functional equation
• History of algebra
• Inequality
• Inequation
• List of equations
• List of scientific equations named after people
• Term (logic)
• Theory of equations

4.9 References
[1] .
[2] “A statement of equality between two expressions. Equations are of two types, identities and conditional equations
(or usually simply “equations”)". « Equation », in Mathematics Dictionary, Glenn James et Robert C. James (éd.), Van
Nostrand, 1968, 3 ed. 1st ed. 1948, p. 131.
[3] The subject of this article is basic in mathematics, and is treated in a lot of textbooks. Among them, Lay 2005, Meyer
2001, and Strang 2005 contain the material of this article.
[4] Thomas, George B., and Finney, Ross L., Calculus and Analytic Geometry, Addison Wesley Publishing Co., fifth edition,
1979, p. 91.
[5] Weisstein, Eric W. “Parametric Equations.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.
com/ParametricEquations.html

26

CHAPTER 4. EQUATION

4.10 External links
• Winplot: General Purpose plotter which can draw and animate 2D and 3D mathematical equations.
• Mathematical equation plotter: Plots 2D mathematical equations, computes integrals, and finds solutions online.
• Equation plotter: A web page for producing and downloading pdf or postscript plots of the solution sets to
equations and inequations in two variables (x and y).
• EqWorld—contains information on solutions to many different classes of mathematical equations.
• fxSolver: Online formula database and graphing calculator for mathematics,natural science and engineering.
• EquationSolver: A webpage that can solve single equations and linear equation systems.
• vCalc: A webpage with an extensive user modifiable equation library.

Chapter 5

Geometric progression

Diagram illustrating three basic geometric sequences of the pattern 1(rn−1 ) up to 6 iterations deep. The first block is a unit block and
the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3
respectively.

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where
each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common
ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5,
2.5, 1.25, ... is a geometric sequence with common ratio 1/2.
Examples of a geometric sequence are powers rk of a fixed number r, such as 2k and 3k . The general form of a
geometric sequence is

a, ar, ar2 , ar3 , ar4 , . . .
where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence’s start value.
27

28

CHAPTER 5. GEOMETRIC PROGRESSION

5.1 Elementary properties
The n-th term of a geometric sequence with initial value a and common ratio r is given by

an = a rn−1 .
Such a geometric sequence also follows the recursive relation
an = r an−1 for every integer n ≥ 1.
Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the
sequence all have the same ratio.
The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers
switching from positive to negative and back. For instance
1, −3, 9, −27, 81, −243, ...
is a geometric sequence with common ratio −3.
The behaviour of a geometric sequence depends on the value of the common ratio.
If the common ratio is:
• Positive, the terms will all be the same sign as the initial term.
• Negative, the terms will alternate between positive and negative.
• Greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of
the initial term).
• 1, the progression is a constant sequence.
• Between −1 and 1 but not zero, there will be exponential decay towards zero.
• −1, the progression is an alternating sequence
• Less than −1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign.
Geometric sequences (with common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as
opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common
difference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population.
Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a
geometric progression, while taking the logarithm of each term in a geometric progression with a positive common
ratio yields an arithmetic progression.
An interesting result of the definition of a geometric progression is that for any value of the common ratio, any three
consecutive terms a, b and c will satisfy the following equation:

b2 = ac
where b is considered to be the geometric mean between a and c.

5.2. GEOMETRIC SERIES

29

5.2 Geometric series
Computation of the sum 2 + 10 + 50 + 250. The sequence is multiplied term by term by 5, and then subtracted from
the original sequence. Two terms remain: the first term, a, and the term one beyond the last, or arm . The desired
result, 312, is found by subtracting these two terms and dividing by 1 − 5.
A geometric series is the sum of the numbers in a geometric progression. For example:

2 + 10 + 50 + 250 = 2 + 2 × 5 + 2 × 52 + 2 × 53 .
Letting a be the first term (here 2), m be the number of terms (here 4), and r be the constant that each term is
multiplied by to get the next term (here 5), the sum is given by:
a(1 − rm )
1−r
In the example above, this gives:

2 + 10 + 50 + 250 =

2(1 − 54 )
−1248
=
= 312.
1−5
−4

The formula works for any real numbers a and r (except r = 1, which results in a division by zero). For example:

−2π + 4π 2 − 8π 3 = −2π + (−2π)2 + (−2π)3 =

5.2.1

−2π(1 − (−2π)3 )
−2π(1 + 8π 3 )
=
≈ −214.855.
1 − (−2π)
1 + 2π

Derivation

To derive this formula, first write a general geometric series as:
n
∑

ark−1 = ar0 + ar1 + ar2 + ar3 + · · · + arn−1 .

k=1

We can find a simpler formula for this sum by multiplying both sides of the above equation by 1 − r, and we'll see
that

(1 − r)

n
∑

ark−1 = (1 − r)(ar0 + ar1 + ar2 + ar3 + · · · + arn−1 )

k=1

= ar0 + ar1 + ar2 + ar3 + · · · + arn−1 − ar1 − ar2 − ar3 − · · · − arn−1 − arn
= a − arn
since all the other terms cancel. If r ≠ 1, we can rearrange the above to get the convenient formula for a geometric
series that computes the sum of n terms:
n
∑

ark−1 =

k=1

5.2.2

a(1 − rn )
.
1−r

Related formulas

If one were to begin the sum not from k=0, but from a different value, say m, then

30

CHAPTER 5. GEOMETRIC PROGRESSION

n
∑

ark =

k=m

a(rm − rn+1 )
.
1−r

Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form
n
∑

ks rk .

k=0

For example:
n
n
d ∑ k ∑ k−1
(n + 1)rn
1 − rn+1
r =
kr
=
−
.
2
dr
(1 − r)
1−r
k=0

k=1

For a geometric series containing only even powers of r multiply by 1 − r2 :

(1 − r2 )

n
∑

ar2k = a − ar2n+2 .

k=0

Then
n
∑

ar2k =

k=0

a(1 − r2n+2 )
.
1 − r2

Equivalently, take r2 as the common ratio and use the standard formulation.
For a series with only odd powers of r

(1 − r2 )

n
∑

ar2k+1 = ar − ar2n+3

k=0

and
n
∑

ar2k+1 =

k=0

5.2.3

ar(1 − r2n+2 )
.
1 − r2

Infinite geometric series

Main article: Geometric series
An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges
if and only if the absolute value of the common ratio is less than one (|r| < 1). Its value can then be computed from
the finite sum formulae
∞
∑

ark = lim

k=0

Since:

n→∞

n
∑
k=0

a
arn+1
a(1 − rn+1 )
=
− lim
n→∞
1−r
1 − r n→∞ 1 − r

ark = lim

5.2. GEOMETRIC SERIES

31

1/8
1/128

1

1/16

1/64

1/4

1/32

1/2
Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 +

which converges to 2.

rn+1 → 0 as n → ∞ when |r| < 1.
Then:
∞
∑

ark =

k=0

a
a
−0=
1−r
1−r

For a series containing only even powers of r ,
∞
∑

ar2k =

k=0

a
1 − r2

and for odd powers only,
∞
∑

ar2k+1 =

k=0

ar
1 − r2

In cases where the sum does not start at k = 0,
∞
∑

ark =

k=m

arm
1−r

The formulae given above are valid only for |r| < 1. The latter formula is valid in every Banach algebra, as long as the
norm of r is less than one, and also in the field of p-adic numbers if |r|p < 1. As in the case for a finite sum, we can
differentiate to calculate formulae for related sums. For example,
∞
∞
1
d ∑ k ∑ k−1
r =
kr
=
dr
(1 − r)2
k=0

k=0

This formula only works for |r| < 1 as well. From this, it follows that, for |r| < 1,

32

CHAPTER 5. GEOMETRIC PROGRESSION

(
)
∞
r 1 + 4r + r2
r (1 + r) ∑ 3 k
kr =
k r =
k r =
2 ;
3 ;
4
(1 − r) k=0
(1 − r) k=0
(1 − r)
k=0
∞
∑

∞
∑

r

k

2 k

Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is an elementary example of a series that converges absolutely.
It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is

1 1 1
1
1/2
+ + +
+ ··· =
= 1.
2 4 8 16
1 − (+1/2)
The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that
converges absolutely.
It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is

1 1 1
1
1/2
1
− + −
+ ··· =
= .
2 4 8 16
1 − (−1/2)
3

5.2.4

Complex numbers

The summation formula for geometric series remains valid even when the common ratio is a complex number. In
this case the condition that the absolute value of r be less than 1 becomes that the modulus of r be less than 1. It is
possible to calculate the sums of some non-obvious geometric series. For example, consider the proposition
∞
∑
sin(kx)
k=0

rk

=

r sin(x)
1 + r2 − 2r cos(x)

The proof of this comes from the fact that

sin(kx) =

eikx − e−ikx
,
2i

which is a consequence of Euler’s formula. Substituting this into the original series gives

∞
∑
sin(kx)
k=0

rk

[ ∞ ( )k
]
∞ ( −ix )k
∑
1 ∑ eix
e
=
−
2i
r
r
k=0

k=0

This is the difference of two geometric series, and so it is a straightforward application of the formula for infinite
geometric series that completes the proof.

5.3 Product
The product of a geometric progression is the product of all terms. If all terms are positive, then it can be quickly
computed by taking the geometric mean of the progression’s first and last term, and raising that mean to the power
given by the number of terms. (This is very similar to the formula for the sum of terms of an arithmetic sequence:
take the arithmetic mean of the first and last term and multiply with the number of terms.)
∏n
i=0

ari =

(√

a0 · an

)n+1

(if a, r > 0 ).

5.4. RELATIONSHIP TO GEOMETRY AND EUCLID’S WORK

33

Proof:
Let the product be represented by P:

P = a · ar · ar2 · · · arn−1 · arn
Now, carrying out the multiplications, we conclude that

P = an+1 r1+2+3+···+(n−1)+(n)
Applying the sum of arithmetic series, the expression will yield

P = an+1 r

n(n+1)
2

n

P = (ar 2 )n+1
We raise both sides to the second power:

P 2 = (a2 rn )n+1 = (a · arn )n+1
Consequently

P 2 = (a0 · an )n+1
P = (a0 · an )

n+1
2

which concludes the proof.

5.4 Relationship to geometry and Euclid’s work
Books VIII and IX of Euclid's Elements analyzes geometric progressions (such as the powers of two, see the article
for details) and give several of their properties.[1]

5.5 See also
• Arithmetic progression
• Arithmetico-geometric sequence
• Exponential function
• Harmonic progression
• Harmonic series
• Infinite series
• Preferred number
• Thomas Robert Malthus
• Geometric distribution

34

CHAPTER 5. GEOMETRIC PROGRESSION

5.6 References
[1]

• Heath, Thomas L. (1956). The Thirteen Books of Euclid’s Elements (2nd ed. [Facsimile. Original publication:
Cambridge University Press, 1925] ed.). New York: Dover Publications.

• Hall & Knight, Higher Algebra, p. 39, ISBN 81-8116-000-2

5.7 External links
• Hazewinkel, Michiel, ed. (2001), “Geometric progression”, Encyclopedia of Mathematics, Springer, ISBN
978-1-55608-010-4
• Derivation of formulas for sum of finite and infinite geometric progression at Mathalino.com
• Geometric Progression Calculator
• Nice Proof of a Geometric Progression Sum at sputsoft.com
• Weisstein, Eric W., “Geometric Series”, MathWorld.

Chapter 6

Geometric series
This article is about infinite geometric series. For finite sums, see geometric progression.
In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the

1/8
1/8

1/4
1/4
1/2

1/2
Each of the purple squares has 1/4 of the area of the next larger square (1/2×1/2 = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the
areas of the purple squares is one third of the area of the large square.

series
35

36

CHAPTER 6. GEOMETRIC SERIES

1
1
1
1
+
+
+
+ ···
2
4
8
16
is geometric, because each successive term can be obtained by multiplying the previous term by 1/2.
Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have
this property. Historically, geometric series played an important role in the early development of calculus, and they
continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and
they have important applications in physics, engineering, biology, economics, computer science, queueing theory,
and finance.

6.1 Common ratio

The convergence of the geometric series with r=1/2 and a=1/2

1/2

1
1/8
1/4
1/32
1/16

The convergence of the geometric series with r=1/2 and a=1

The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series
is constant. This relationship allows for the representation of a geometric series using only two terms, r and a. The
term r is the common ratio, and a is the first term of the series. As an example the geometric series given in the
introduction,
1
2

+

1
4

+

1
8

+

1
16

+ ···

6.2. SUM

37

may simply be written as
a + ar + ar2 + ar3 + · · · , with r =

1
2

and a =

1
2

.

The following table shows several geometric series with different common ratios:
The behavior of the terms depends on the common ratio r:
If r is between −1 and +1, the terms of the series become smaller and smaller, approaching zero in the
limit and the series converges to a sum. In the case above, where r is one half, the series has the sum
one.
If r is greater than one or less than minus one the terms of the series become larger and larger in
magnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The series
diverges.)
If r is equal to one, all of the terms of the series are the same. The series diverges.
If r is minus one the terms take two values alternately (e.g. 2, −2, 2, −2, 2,... ). The sum of the terms
oscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again the
series has no sum. See for example Grandi’s series: 1 − 1 + 1 − 1 + ···.

6.2 Sum
The sum of a geometric series is finite as long as the absolute value of the ratio is less than 1; as the numbers near
zero, they become insignificantly small, allowing a sum to be calculated despite the series containing infinitely-many
terms. The sum can be computed using the self-similarity of the series.

6.2.1

Example

A self-similar illustration of the sum s. Removing the largest circle results in a similar figure of 2/3 the original size.

Consider the sum of the following geometric series:

s = 1+

4
8
2
+
+
+ ···
3
9
27

This series has common ratio 2/3. If we multiply through by this common ratio, then the initial 1 becomes a 2/3, the
2/3 becomes a 4/9, and so on:

2
2
4
8
16
s =
+
+
+
+ ···
3
3
9
27
81

38

CHAPTER 6. GEOMETRIC SERIES

This new series is the same as the original, except that the first term is missing. Subtracting the new series (2/3)s from
the original series s cancels every term in the original but the first:

s−

2
s = 1, so s = 3.
3

A similar technique can be used to evaluate any self-similar expression.

6.2.2

Formula

For r ̸= 1 , the sum of the first n terms of a geometric series is:

a + ar + ar2 + ar3 + · · · + arn−1 =

n−1
∑

ark = a

k=0

1 − rn
,
1−r

where a is the first term of the series, and r is the common ratio. We can derive this formula as follows:
Lets = a + ar + ar2 + ar3 + · · · + arn−1 .
Thenrs = ar + ar2 + ar3 + ar4 + · · · + arn
Thens − rs = a − arn
Thens(1 − r) = a(1 − rn ), so s = a

1 − rn
1−r

(ifr ̸= 1).

As n goes to infinity, the absolute value of r must be less than one for the series to converge. The sum then becomes

a + ar + ar2 + ar3 + ar4 + · · · =

∞
∑

ark =

k=0

a
, for |r| < 1.
1−r

When a = 1, this can be simplified to:

1 + r + r2 + r3 + · · · =

1
,
1−r

the left-hand side being a geometric series with common ratio r. We can derive this formula:
Lets = 1 + r + r2 + r3 + · · · .
Thenrs = r + r2 + r3 + · · · .
Thens − rs = 1, so s(1 − r) = 1, thus and s =

1
.
1−r

The general formula follows if we multiply through by a.
The formula holds true for complex “r”, with the same restrictions (modulus of “r” is strictly less than one).

6.2.3

Proof of convergence

We can prove that the geometric series converges using the sum formula for a geometric progression:
1 + r + r2 + r3 + · · · = lim

n→∞

(
)
1 + r + r2 + · · · + rn

1 − rn+1
n→∞
1−r

= lim

6.2. SUM

39

Since (1 + r + r2 + ... + rn )(1−r) = 1−rn+1 and rn+1 → 0 for | r | < 1.
Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series.
Consider the function:

rK
1−r

g(K) =

Note that:

1 = g(0) − g(1), r = g(1) − g(2), r2 = g(2) − g(3), · · ·
Thus:

S = 1 + r + r2 + r3 + ... = (g(0) − g(1)) + (g(1) − g(2)) + (g(2) − g(3)) + · · ·
If

|r| < 1
then

g(K) −→ 0 as K → ∞
So S converges to

g(0) =

1
.
1−r

6.2.4

Generalized formula

For r ̸= 1 , the sum of the first n terms of a geometric series is:
b
∑

rk =

k=a

ra − rb+1
,
1−r

where a, b ∈ N .
We can derive this formula as follows:
we put b = n − 1 ⇒ n = b + 1
b
∑
k=a

rk =

n−1
∑
k=0

rk −

a−1
∑

rk

k=0

1 − rn
1 − ra
=
−
1−r
1−r
1 − rn − 1 + ra
=
1−r
a
r − rb+1
=
1−r

40

CHAPTER 6. GEOMETRIC SERIES

6.3 Applications
6.3.1

Repeating decimals

Main article: Repeating decimal
A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:

0.7777 . . . =

7
7
7
7
+
+
+
+ ··· .
10
100
1000
10000

The formula for the sum of a geometric series can be used to convert the decimal to a fraction:

0.7777 . . . =

a
7/10
7
=
= .
1−r
1 − 1/10
9

The formula works not only for a single repeating figure, but also for a repeating group of figures. For example:

0.123412341234 . . . =

a
1234/10000
1234
=
=
.
1−r
1 − 1/10000
9999

Note that every series of repeating consecutive decimals can be conveniently simplified with the following:

0.09090909 . . . =

09
1
=
.
99
11

0.143814381438 . . . =

1438
.
9999

9
= 1.
9
That is, a repeating decimal with repeat length n is equal to the quotient of the repeating part (as an integer) and 10n
- 1.
0.9999 . . . =

6.3.2

Archimedes’ quadrature of the parabola

Main article: The Quadrature of the Parabola
Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His
method was to dissect the area into an infinite number of triangles.
Archimedes’ Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.
Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the
area of a green triangle, and so forth.
Assuming that the blue triangle has area 1, the total area is an infinite sum:

1+2

( )2
( )3
( )
1
1
1
+4
+8
+ ··· .
8
8
8

The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third
term the areas of the four yellow triangles, and so on. Simplifying the fractions gives

1+

1
1
1
+
+
+ ··· .
4
16
64

6.3. APPLICATIONS

41

Archimedes’ dissection of a parabolic segment into infinitely many triangles

This is a geometric series with common ratio 1/4 and the fractional part is equal to
∞
∑

4−n = 1 + 4−1 + 4−2 + 4−3 + · · · =

n=0

4
.
3

The sum is

1
1
=
1−r
1−

1
4

=

4
.
3

This computation uses the method of exhaustion, an early version of integration. In modern calculus, the same area
could be found using a definite integral.

6.3.3

Fractal geometry

In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure.
For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles
(see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore
has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the
blue triangle as a unit of area, the total area of the snowflake is
( )2
( )3
( )
1
1
1
+ 12
+ 48
+ ··· .
1+3
9
9
9
The first term of this series represents the area of the blue triangle, the second term the total area of the three green
triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series
is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is

42

CHAPTER 6. GEOMETRIC SERIES

The interior of the Koch snowflake is a union of infinitely many triangles.

1+

1
a
= 1+ 3
1−r
1−

4
9

=

8
.
5

Thus the Koch snowflake has 8/5 of the area of the base triangle.

6.3.4

Zeno’s paradoxes

Main article: Zeno’s paradoxes
The convergence of a geometric series reveals that a sum involving an infinite number of summands can indeed be
finite, and so allows one to resolve many of Zeno's paradoxes. For example, Zeno’s dichotomy paradox maintains

6.3. APPLICATIONS

43

that movement is impossible, as one can divide any finite path into an infinite number of steps wherein each step is
taken to be half the remaining distance. Zeno’s mistake is in the assumption that the sum of an infinite number of
finite steps cannot be finite. This is of course not true, as evidenced by the convergence of the geometric series with
r = 1/2 .

6.3.5

Euclid

Book IX, Proposition 35[1] of Euclid’s Elements expresses the partial sum of a geometric series in terms of members
of the series. It is equivalent to the modern formula.

6.3.6

Economics

Main article: Time value of money
In economics, geometric series are used to represent the present value of an annuity (a sum of money to be paid in
regular intervals).
For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end of
the year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannot
invest the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + I
), where I is the yearly interest rate.
Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + I )2 (squared because two
years’ worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100
per year in perpetuity is
∞
∑

$100
,
(1
+ I)n
n=1
which is the infinite series:

$100
$100
$100
$100
+
+
+
+ ··· .
(1 + I)
(1 + I)2
(1 + I)3
(1 + I)4
This is a geometric series with common ratio 1 / (1 + I ). The sum is the first term divided by (one minus the common
ratio):

$100/(1 + I)
$100
=
.
1 − 1/(1 + I)
I
For example, if the yearly interest rate is 10% ( I = 0.10), then the entire annuity has a present value of $100 / 0.10
= $1000.
This sort of calculation is used to compute the APR of a loan (such as a mortgage loan). It can also be used to estimate
the present value of expected stock dividends, or the terminal value of a security.

6.3.7

Geometric power series

The formula for a geometric series
1
= 1 + x + x2 + x3 + x4 + · · ·
1−x
can be interpreted as a power series in the Taylor’s theorem sense, converging where |x| < 1 . From this, one can
extrapolate to obtain other power series. For example,

44

CHAPTER 6. GEOMETRIC SERIES

∫

dx
1 + x2
∫
dx
=
1 − (−x2 )
∫ (
)
(
) (
)2 (
)3
=
1 + −x2 + −x2 + −x2 + · · · dx
∫
(
)
=
1 − x2 + x4 − x6 + · · · dx

tan−1 (x) =

x3
x5
x7
+
−
+ ···
3
5
7
∞
∑ (−1)n
x2n+1
=
2n
+
1
n=0

=x−

By differentiating the geometric series, one obtains the variant[2]
∞
∑

nxn−1 =

n=1

1
(1 − x)2

for |x| < 1.

Similarly obtained are:
∞
∑

n(n − 1)xn−2 =

n=2
∞
∑

2
(1 − x)3

n(n − 1)(n − 2)xn−3 =

n=3

for |x| < 1,

6
(1 − x)4

for |x| < 1.

6.4 See also
• 0.999...
• Asymptote
• Divergent geometric series
• Generalized hypergeometric function
• Geometric progression
• Neumann series
• Ratio test
• Root test
• Series (mathematics)
• Tower of Hanoi

6.4.1

Specific geometric series

• Grandi’s series: 1 − 1 + 1 − 1 + ⋯
• 1+2+4+8+⋯
• 1−2+4−8+⋯

6.5. REFERENCES

45

• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯

6.5 References
[1] “Euclid’s Elements, Book IX, Proposition 35”. Aleph0.clarku.edu. Retrieved 2013-08-01.
[2] Taylor, Angus E. (1955), Advanced Calculus, Blaisdell, p. 603

• Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.
• Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 278-279, 1985.
• Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987.
• Courant, R. and Robbins, H. “The Geometric Progression.” §1.2.3 in What Is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 13-14, 1996.
• Pappas, T. “Perimeter, Area & the Infinite Series.” The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, pp. 134-135, 1989.
• James Stewart (2002). Calculus, 5th ed., Brooks Cole. ISBN 978-0-534-39339-7
• Larson, Hostetler, and Edwards (2005). Calculus with Analytic Geometry, 8th ed., Houghton Mifflin Company.
ISBN 978-0-618-50298-1
• Roger B. Nelsen (1997). Proofs without Words: Exercises in Visual Thinking, The Mathematical Association
of America. ISBN 978-0-88385-700-7
• Andrews, George E. (1998). “The geometric series in calculus”. The American Mathematical Monthly (Mathematical Association of America) 105 (1): 36–40. doi:10.2307/2589524. JSTOR 2589524.

6.5.1

History and philosophy

• C. H. Edwards, Jr. (1994). The Historical Development of the Calculus, 3rd ed., Springer. ISBN 978-0-38794313-8.
• Swain, Gordon and Thomas Dence (April 1998). “Archimedes’ Quadrature of the Parabola Revisited”. Mathematics Magazine 71 (2): 123–30. doi:10.2307/2691014. JSTOR 2691014.
• Eli Maor (1991). To Infinity and Beyond: A Cultural History of the Infinite, Princeton University Press. ISBN
978-0-691-02511-7
• Morr Lazerowitz (2000). The Structure of Metaphysics (International Library of Philosophy), Routledge. ISBN
978-0-415-22526-7

6.5.2

Economics

• Carl P. Simon and Lawrence Blume (1994). Mathematics for Economists, W. W. Norton & Company. ISBN
978-0-393-95733-4
• Mike Rosser (2003). Basic Mathematics for Economists, 2nd ed., Routledge. ISBN 978-0-415-26784-7

46

CHAPTER 6. GEOMETRIC SERIES

6.5.3

Biology

• Edward Batschelet (1992). Introduction to Mathematics for Life Scientists, 3rd ed., Springer. ISBN 978-0-38709648-3
• Richard F. Burton (1998). Biology by Numbers: An Encouragement to Quantitative Thinking, Cambridge University Press. ISBN 978-0-521-57698-7

6.5.4

Computer science

• John Rast Hubbard (2000). Schaum’s Outline of Theory and Problems of Data Structures With Java, McGrawHill. ISBN 978-0-07-137870-3

6.6 External links
• Hazewinkel, Michiel, ed. (2001), “Geometric progression”, Encyclopedia of Mathematics, Springer, ISBN
978-1-55608-010-4
• Weisstein, Eric W., “Geometric Series”, MathWorld.
• Geometric Series at PlanetMath.org.
• Peppard, Kim. “College Algebra Tutorial on Geometric Sequences and Series”. West Texas A&M University.
• Casselman, Bill. “A Geometric Interpretation of the Geometric Series” (Applet).
• “Geometric Series” by Michael Schreiber, Wolfram Demonstrations Project, 2007.

Chapter 7

Scale factor
A scale factor is a number which scales, or multiplies, some quantity. In the equation y = Cx, C is the scale factor
for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x. For example, doubling
distances corresponds to a scale factor of two for distance, while cutting a cake in half results in pieces with a scale
factor of one half. The basic equation for it is image over preimage.
In the field of measurements, the scale factor of an instrument is sometimes referred to as sensitivity. The ratio of
any two corresponding lengths in two similar geometric figures is also called a scale factor.

7.1 See also
• Scale (ratio)
• Scale (map)
• Scales of scale models
• Scaling (geometry)
• Scalar (mathematics)
• Scaling in gravity
• Scaling in statistical estimation
• Scale factor (computer science)
• Scale factor (cosmology)
• Orthogonal coordinates

47

Chapter 8

Sequence
“Sequential” redirects here. For the manual transmission, see Sequential manual transmission. For other uses, see
Sequence (disambiguation).
In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed. Like a set, it contains
members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the
sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions
in the sequence. Formally, a sequence can be defined as a function whose domain is a countable totally ordered set,
such as the natural numbers.
For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from
(A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a
valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive
integers (2, 4, 6,...). In computing and computer science, finite sequences are sometimes called strings, words or lists,
the different names commonly corresponding to different ways to represent them into computer memory; infinite
sequences are also called streams. The empty sequence ( ) is included in most notions of sequence, but may be
excluded depending on the context.

An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. It
is, however, bounded.

48

8.1. EXAMPLES AND NOTATION

49

8.1 Examples and notation
A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of
mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence
properties of sequences. In particular, sequences are the basis for series, which are important in differential equations
and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the
study of prime numbers.
There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences.
One way to specify a sequence is to list the elements. For example, the first four odd numbers form the sequence
(1,3,5,7). This notation can be used for infinite sequences as well. For instance, the infinite sequence of positive
odd integers can be written (1,3,5,7,...). Listing is most useful for infinite sequences with a pattern that can be easily
discerned from the first few elements. Other ways to denote a sequence are discussed after the examples.

8.1.1

Important examples

2

3

1 1
8
5

A tiling with squares whose sides are successive Fibonacci numbers in length.

There are many important integer sequences. The prime numbers are the natural numbers bigger than 1, that have
no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2,3,5,7,11,13,17,...). The
study of prime numbers has important applications for mathematics and specifically number theory.
The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. The first
two elements are either 0 and 1 or 1 and 1 so that the sequence is (0,1,1,2,3,5,8,13,21,34,...).
Other interesting sequences include the ban numbers, whose spellings do not contain a certain letter of the alphabet.
For instance, the eban numbers (do not contain 'e') form the sequence (2,4,6,30,32,34,36,40,42,...). Another sequence
based on the English spelling of the letters is the one based on their number of letters (3,3,5,4,4,3,5,5,4,3,6,6,8,...).
For a list of important examples of integers sequences see On-line Encyclopedia of Integer Sequences.
Other important examples of sequences include ones made up of rational numbers, real numbers, and complex numbers. The sequence (.9,.99,.999,.9999,...) approaches the number 1. In fact, every real number can be written as
the limit of a sequence of rational numbers. It is this fact that allows us to write any real number as the limit of a
sequence of decimals. For instance, π is the limit of the sequence (3,3.1,3.14,3.141,3.1415,...). The sequence for π,
however, does not have any pattern that is easily discernible by eye, unlike the sequence (0.9,0.99,...).

50

CHAPTER 8. SEQUENCE

8.1.2

Indexing

Other notations can be useful for sequences whose pattern cannot be easily guessed, or for sequences that do not have
a pattern such as the digits of π. This section focuses on the notations used for sequences that are a map from a subset
of the natural numbers. For generalizations to other countable index sets see the following section and below.
The terms of a sequence are commonly denoted by a single variable, say an, where the index n indicates the nth
element of the sequence.
a1 ↔
a2 ↔
a3 ↔
..
.
an−1 ↔

element 1st
element 2nd
element 3rd
..
.
element (n-1)th

an ↔
an+1 ↔
..
.

element nth
element (n+1)th
..
.

Indexing notation is used to refer to a sequence in the abstract. It is also a natural notation for sequences whose
elements are related to the index n (the element’s position) in a simple way. For instance, the sequence of the first 10
square numbers could be written as

(a1 , a2 , ..., a10 ),

ak = k 2 .

This represents the sequence (1,4,9,...100). This notation is often simplified further as

(ak )10
k=1 ,

ak = k 2 .

Here the subscript {k=1} and superscript 10 together tell us that the elements of this sequence are the ak such that k
= 1, 2, ..., 10.
Sequences can be indexed beginning and ending from any integer. The infinity symbol ∞ is often used as the superscript to indicate the sequence including all integer k-values starting with a certain one. The sequence of all positive
squares is then denoted

(ak )∞
k=1 ,

ak = k 2 .

In cases where the set of indexing numbers is understood, such as in analysis, the subscripts and superscripts are often
left off. That is, one simply writes ak for an arbitrary sequence. In analysis, k would be understood to run from 1 to
∞. However, sequences are often indexed starting from zero, as in

(ak )∞
k=0 = (a0 , a1 , a2 , ...).
In some cases the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily
inferred. In these cases the index set may be implied by a listing of the first few abstract elements. For instance, the
sequence of squares of odd numbers could be denoted in any of the following ways.
• (1, 9, 25, ...)
• (a1 , a3 , a5 , ...),
• (a2k−1 )∞
k=1 ,

ak = k 2
ak = k 2

8.2. FORMAL DEFINITION AND BASIC PROPERTIES
• (ak )∞
k=1 ,

51

ak = (2k − 1)2

• ((2k − 1)2 )∞
k=1
Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations if the
indexing set was understood to be the natural numbers.
Finally, sequences can more generally be denoted by writing a set inclusion in the subscript, such as in

(ak )k∈N
The set of values that the index can take on is called the index set. In general, the ordering of the elements ak is
specified by the order of the elements in the indexing set. When N is the index set, the element ak+1 comes after the
element ak since in N, the element (k+1) comes directly after the element k.

8.1.3

Specifying a sequence by recursion

Sequences whose elements are related to the previous elements in a straightforward way are often specified using
recursion. This is in contrast to the specification of sequence elements in terms of their position.
To specify a sequence by recursion requires a rule to construct each consecutive element in terms of the ones before
it. In addition, enough initial elements must be specified so that new elements of the sequence can be specified by
the rule. The principle of mathematical induction can be used to prove that a sequence is well-defined, which is to
say that that every element of the sequence is specified at least once and has a single, unambiguous value. Induction
can also be used to prove properties about a sequence, especially for sequences whose most natural specification is
by recursion.
The Fibonacci sequence can be defined using a recursive rule along with two initial elements. The rule is that each
element is the sum of the previous two elements, and the first two elements are 0 and 1.
an = an−1 + an−2 , with a0 = 0 and a1 = 1 .
The first ten terms of this sequence are 0,1,1,2,3,5,8,13,21, and 34. A more complicated example of a sequence that
is defined recursively is Recaman’s sequence, considered at the beginning of this section. We can define Recaman’s
sequence by
a0 = 0 and an = an−1 −n if the result is positive and not already in the list. Otherwise, an = an−1 +n
.
Not all sequences can be specified by a rule in the form of an equation, recursive or not, and some can be quite
complicated. For example, the sequence of prime numbers is the set of prime numbers in their natural order. This
gives the sequence (2,3,5,7,11,13,17,...).
One can also notice that the next element of a sequence is a function of the element before, and so we can write the
next element as : an+1 = f (an )
This functional notation can prove useful when one wants to prove the global monotony of the sequence.

8.2 Formal definition and basic properties
There are many different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered
by the definitions and notations introduced below.

8.2.1

Formal definition

A sequence is usually defined as a function whose domain is a countable totally ordered set, although in many disciplines the domain is restricted, such as to the natural numbers. In real analysis a sequence is a function from a subset

52

CHAPTER 8. SEQUENCE

of the natural numbers to the real numbers.[1] In other words, a sequence is a map f(n) : N → R. To recover our
earlier notation we might identify an = f(n) for all n or just write an : N → R.
In complex analysis, sequences are defined as maps from the natural numbers to the complex numbers (C).[2] In
topology, sequences are often defined as functions from a subset of the natural numbers to a topological space.[3]
Sequences are an important concept for studying functions and, in topology, topological spaces. An important generalization of sequences, called a net, is to functions from a (possibly uncountable) directed set to a topological space.

8.2.2

Finite and infinite

The length of a sequence is defined as the number of terms in the sequence.
A sequence of a finite length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has no
elements.
Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other—the
sequence has a first element, but no final element, it is called a singly infinite, or one-sided (infinite) sequence,
when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a
first nor a final element—is called a bi-infinite sequence, two-way infinite sequence, or doubly infinite sequence.
A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( …, −4, −2,
0, 2, 4, 6, 8… ), is bi-infinite. This sequence could be denoted (2n)∞
n=−∞ .
One can interpret singly infinite sequences as elements of the semigroup ring of the natural numbers R[N], and doubly
infinite sequences as elements of the group ring of the integers R[Z]. This perspective is used in the Cauchy product
of sequences.

8.2.3

Increasing and decreasing

A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For a
sequence (an )∞
n=1 this can be written as an ≤ an₊₁ for all n ∈ N. If each consecutive term is strictly greater than
(>) the previous term then the sequence is called strictly monotonically increasing. A sequence is monotonically
decreasing if each consecutive term is less than or equal to the previous one, and strictly monotonically decreasing
if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone
sequence. This is a special case of the more general notion of a monotonic function.
The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid
any possible confusion with strictly increasing and strictly decreasing, respectively.

8.2.4

Bounded

If the sequence of real numbers (an) is such that all the terms, after a certain one, are less than some real number M,
then the sequence is said to be bounded from above. In less words, this means an ≤ M for all n greater than N for
some pair M and N. Any such M is called an upper bound. Likewise, if, for some real m, an ≥ m for all n greater
than some N, then the sequence is bounded from below and any such m is called a lower bound. If a sequence is
both bounded from above and bounded from below then the sequence is said to be bounded.

8.2.5

Other types of sequences

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements
without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even
integers (2,4,6,...) is a subsequence of the positive integers (1,2,3,...). The positions of some elements change when
other elements are deleted. However, the relative positions are preserved.
Some other types of sequences that are easy to define include:
• An integer sequence is a sequence whose terms are integers.
• A polynomial sequence is a sequence whose terms are polynomials.

8.3. LIMITS AND CONVERGENCE

53

• A positive integer sequence is sometimes called multiplicative if anm = an am for all pairs n,m such that n and
m are coprime.[4] In other instances, sequences are often called multiplicative if an = na1 for all n. Moreover,
the multiplicative Fibonacci sequence satisfies the recursion relation an = an₋₁ an₋₂.

8.3 Limits and convergence
Main article: Limit of a sequence
One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit.

1.0
0.8
0.6
0.4

n+1
2n2

0.2
0.0
5

10

15

20

25

The plot of a convergent sequence (a ) is shown in blue. Visually we can see that the sequence is converging to the limit zero as n
increases.

Continuing informally, a (singly infinite) sequence has a limit if it approaches some value L, called the limit, as n
becomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the value
of an approaches L as n → ∞, denoted

lim an = L.

n→∞

More precisely, the sequence converges if there exists a limit L such that the remaining a 's are arbitrarily close to L
for some n large enough.
If a sequence converges to some limit, then it is convergent; otherwise it is divergent.
If an gets arbitrarily large as n → ∞ we write

lim an = ∞.

n→∞

54

CHAPTER 8. SEQUENCE

In this case we say that the sequence (an) diverges, or that it converges to infinity.
If an becomes arbitrarily “small” negative numbers (large in magnitude) as n → ∞ we write

lim an = −∞

n→∞

and say that the sequence diverges or converges to minus infinity.

8.3.1

Definition of convergence

For sequences that can be written as (an )∞
n=1 with an ∈ R we can write (an) with the indexing set understood as N.
These sequences are most common in real analysis. The generalizations to other types of sequences are considered
in the following section and the main page Limit of a sequence.
Let (an) be a sequence. In words, the sequence (an) is said to converge if there exists a number L such that no matter
how close we want the an to be to L (say ε-close where ε > 0), we can find a natural number N such that all terms
(aN+1, aN+2, ...) are further closer to L (within ε of L). [1] This is often written more compactly using symbols. For
instance,
for all ε > 0, there exists a natural number N such that L−ε < an < L+ε for all n ≥ N.
In even more compact notation

∀ϵ > 0, ∃N ∈ N s.t. ∀n ≥ N, |an − L| < ϵ.
The difference in the definitions of convergence for (one-sided) sequences in complex
√ analysis and metric spaces is
that the absolute value |an − L| is interpreted as the distance in the complex plane ( z ∗ z ), and the distance under
the appropriate metric, respectively.

8.3.2

Applications and important results

Important results for convergence and limits of (one-sided) sequences of real numbers include the following. These
equalities are all true at least when both sides exist. For a discussion of when the existence of the limit on one side
implies the existence of the other see a real analysis text such as can be found in the references.[1][5]
• The limit of a sequence is unique.
• limn→∞ (an ± bn ) = limn→∞ an ± limn→∞ bn
• limn→∞ can = c limn→∞ an
• limn→∞ (an bn ) = (limn→∞ an )(limn→∞ bn )
• limn→∞

an
bn

=

limn→∞ an
limn→∞ bn

provided limn→∞ bn ̸= 0

• limn→∞ apn = [limn→∞ an ]

p

• If an ≤ bn for all n greater than some N, then limn→∞ an ≤ limn→∞ bn .
• (Squeeze Theorem) If an ≤ cn ≤ bn for all n > N, and limn→∞ an = limn→∞ bn = L , then limn→∞ cn = L
.
• If a sequence is bounded and monotonic then it is convergent.
• A sequence is convergent if and only if every subsequence is convergent.

8.4. SERIES

55

The plot of a Cauchy sequence (X ), shown in blue, as X versus n. Visually, we see that the sequence appears to be converging to the
limit zero as the terms in the sequence become closer together as n increases. In the real numbers every Cauchy sequence converges
to some limit.

8.3.3

Cauchy sequences

Main article: Cauchy sequence
A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of
a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One
particularly important result in real analysis is Cauchy characterization of convergence for sequences:
In the real numbers, a sequence is convergent if and only if it is Cauchy.
In contrast, in the rational numbers, e.g. the sequence defined by x1 = 1 and xn₊₁ = xn + 2/xn/2 is Cauchy, but has no
rational limit, cf. here.

8.4 Series
Main article: Series (mathematics)
A series is, informally speaking, the sum of the terms of a sequence. That is, adding the first N terms of a (one-sided)
sequence forms the Nth term of another sequence, called a series. Thus the N series of the sequence (a ) results in
another sequence (SN) given by:

S1 = a 1
S2 = a1 + a2
S3 = a1 + a2
..
..
.
.
SN = a1 + a2
..
..
.
.

+ a3

+ a3 + · · ·

We can also write the nth term of the series as

56

CHAPTER 8. SEQUENCE

SN =

N
∑

an .

n=1

Then the concepts used to talk about sequences, such as convergence, carry over to series (the sequence of partial
sums) and the properties can be characterized as properties of the underlying sequences (such as (an) in the last
example). The limit, if it exists, of an infinite series (the series created from an infinite sequence) is written as

lim SN =

N →∞

∞
∑

an .

n=1

8.5 Use in other fields of mathematics
8.5.1

Topology

Sequence play an important role in topology, especially in the study of metric spaces. For instance:
• A metric space is compact exactly when it is sequentially compact.
• A function from a metric space to another metric space is continuous exactly when it takes convergent sequences
to convergent sequences.
• A metric space is a connected space if, whenever the space is partitioned into two sets, one of the two sets
contains a sequence converging to a point in the other set.
• A topological space is separable exactly when there is a dense sequence of points.
Sequences can be generalized to nets or filters. These generalizations allow one to extend some of the above theorems
to spaces without metrics.
Product topology
A product space of a sequence of topological spaces is the cartesian product of the spaces equipped with a natural
topology called the product topology.
More formally, given a sequence of spaces {Xi } , define X such that

X :=

∏

Xi ,

i∈I

is the set of sequences {xi } where each xi is an element of Xi . Let the canonical projections be written as pi :
X → Xi. Then the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest
open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff
topology.

8.5.2

Analysis

In analysis, when talking about sequences, one will generally consider sequences of the form

(x1 , x2 , x3 , . . . ) or (x0 , x1 , x2 , . . . )
which is to say, infinite sequences of elements indexed by natural numbers.

8.5. USE IN OTHER FIELDS OF MATHEMATICS

57

It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined
by xn = 1/log(n) would be defined only for n ≥ 2. When talking about such infinite sequences, it is usually sufficient
(and does not change much for most considerations) to assume that the members of the sequence are defined at least
for all indices large enough, that is, greater than some given N.
The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type
can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often
function spaces. Even more generally, one can study sequences with elements in some topological space.
Sequence spaces
Main article: Sequence space
A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it
is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers.
The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K,
and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar
multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with
a norm, or at least the structure of a topological vector space.
The most important sequences spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences,
with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers.
Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively
denoted c and c0 , with the sup norm. Any sequence space can also be equipped with the topology of pointwise
convergence, under which it becomes a special kind of Fréchet space called FK-space.

8.5.3

Linear algebra

Sequences over a field may also be viewed as vectors in a vector space. Specifically, the set of F-valued sequences
(where F is a field) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.

8.5.4

Abstract algebra

Abstract algebra employs several types of sequences, including sequences of mathematical objects such as groups or
rings.
Free monoid
Main article: Free monoid
If A is a set, the free monoid over A (denoted A* , also called Kleene star of A) is a monoid containing all the finite
sequences (or strings) of zero or more elements of A, with the binary operation of concatenation. The free semigroup
A+ is the subsemigroup of A* containing all elements except the empty sequence.
Exact sequences
Main article: Exact sequence
In the context of group theory, a sequence

f1

f2

f3

fn

G0 −→ G1 −→ G2 −→ · · · −→ Gn
of groups and group homomorphisms is called exact if the image (or range) of each homomorphism is equal to the
kernel of the next:

58

CHAPTER 8. SEQUENCE

im(fk ) = ker(fk+1 )
Note that the sequence of groups and homomorphisms may be either finite or infinite.
A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence
of vector spaces and linear maps, or of modules and module homomorphisms.
Spectral sequences
Main article: Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups
by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their
introduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.

8.5.5

Set theory

An ordinal-indexed sequence is a generalization of a sequence. If α is a limit ordinal and X is a set, an α-indexed
sequence of elements of X is a function from α to X. In this terminology an ω-indexed sequence is an ordinary
sequence.

8.5.6

Computing

Automata or finite state machines can typically be thought of as directed graphs, with edges labeled using some specific
alphabet, Σ. Most familiar types of automata transition from state to state by reading input letters from Σ, following
edges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word).
The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministic
automaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some input
letter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence of
single states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally used
to mean the latter.

8.5.7

Streams

Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as sequences or streams, as opposed to finite strings. Infinite binary
sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0, 1}). The set C = {0,
1}∞ of all infinite, binary sequences is sometimes called the Cantor space.
An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to
1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalization
method for proofs.[6]

8.6 Types
• ±1-sequence
• Arithmetic progression
• Cauchy sequence
• Farey sequence
• Fibonacci sequence

8.7. RELATED CONCEPTS

59

• Geometric progression
• Look-and-say sequence
• Thue–Morse sequence

8.7 Related concepts
• List (computing)
• Ordinal-indexed sequence
• Recursion (computer science)
• Tuple
• Set theory

8.8 Operations
• Cauchy product
• Limit of a sequence

8.9 See also
• Enumeration
• Net (topology) (a generalization of sequences)
• On-Line Encyclopedia of Integer Sequences
• Permutation
• Recurrence relation
• Sequence space
• Set (mathematics)

8.10 References
[1] Gaughan, Edward. “1.1 Sequences and Convergence”. Introduction to Analysis. AMS (2009). ISBN 0-8218-4787-2.
[2] Edward B. Saff & Arthur David Snider (2003). “Chapter 2.1”. Fundamentals of Complex Analysis. ISBN 01-390-7874-6.
[3] James R. Munkres. “Chapters 1&2”. Topology. ISBN 01-318-1629-2.
[4] Lando, Sergei K. “7.4 Multiplicative sequences”. Lectures on generating functions. AMS. ISBN 0-8218-3481-9.
[5] Dawikins, Paul. “Series and Sequences”. Paul’s Online Math Notes/Calc II (notes). Retrieved 18 December 2012.
[6] Oflazer, Kemal. “FORMAL LANGUAGES, AUTOMATA AND COMPUTATION: DECIDABILITY” (PDF). cmu.edu.
Carnegie-Mellon University. Retrieved 24 April 2015.

60

CHAPTER 8. SEQUENCE

8.11 External links
• The dictionary definition of sequence at Wiktionary
• Hazewinkel, Michiel, ed. (2001), “Sequence”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608010-4
• The On-Line Encyclopedia of Integer Sequences
• Journal of Integer Sequences (free)
• Sequence at PlanetMath.org.

8.12. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

61

8.12 Text and image sources, contributors, and licenses
8.12.1

Text

• Amplitude Source: https://en.wikipedia.org/wiki/Amplitude?oldid=672427896 Contributors: Mav, Bryan Derksen, Ed Poor, Andre Engels, DrBob, Ryguasu, Michael Hardy, Bewildebeast, Ixfd64, Ahoerstemeier, Angela, Glenn, AugPi, Nikai, Andres, Smack, Pheon,
Hyacinth, Omegatron, Jph, Denelson83, Rogper~enwiki, Donarreiskoffer, Robbot, Merovingian, Wikibot, Giftlite, DavidCary, Art Carlson, TomViza, CryptoDerk, PricklyPear, Icairns, Trevor MacInnis, Discospinster, Vsmith, El C, Alberto Orlandini, Stesmo, Smalljim,
Nk, MPerel, Nsaa, Jason One, Wendell, Alansohn, Enirac Sum, Wtshymanski, SteinbDJ, Chirpy, Kbolino, Oleg Alexandrov, LOL, SeventyThree, Eugeneiiim, Jivecat, The wub, FlaBot, Mathbot, Pevernagie, M7bot, Srleffler, Chobot, Bgwhite, YurikBot, Splash, Zwobot,
Pele Merengue, Kelovy, Alex Ruddick, Katieh5584, KnightRider~enwiki, SmackBot, Hydrogen Iodide, Mirokado, Bluebot, JackyR,
WendelScardua, MalafayaBot, Complexica, Jerome Charles Potts, Octahedron80, DHN-bot~enwiki, Can't sleep, clown will eat me,
Mr.Z-man, Tlusťa, Aldaron, Radagast83, Nakon, Dreadstar, Drhamptn, Yevgeny Kats, Andrei Stroe, Kuru, Mathel, Dr Greg, Xionbox,
JYi, Iridescent, Shoeofdeath, Fapae~enwiki, Tawkerbot2, MightyWarrior, Eastlaw, Switchercat, Verdi1, Sopoforic, Cydebot, Meno25,
Michaelas10, Islander, Quibik, Mr Gronk, Epbr123, Headbomb, Bobblehead, Hmrox, AntiVandalBot, QuiteUnusual, Quintote, JAnDbot, Fetchcomms, Acroterion, WolfmanSF, VoABot II, Just James, Hdt83, MartinBot, PrestonH, J.delanoy, Terrek, Minesweeper.007,
Entropy, Kvdveer, Waterppk, VolkovBot, CWii, ABF, Liverwort, Bluetetrahedron, TXiKiBoT, Technopat, Hqb, Rei-bot, Qxz, Ferengi,
BotKung, Gillyweed, Enviroboy, Spinningspark, Blaaake, Dmcq, AlleborgoBot, BotMultichill, Mungo Kitsch, Happysailor, Prestonmag,
Freezeframehigh5, Dabomb87, WikipedianMarlith, ClueBot, The Thing That Should Not Be, Cptmurdok, Mild Bill Hiccup, Blanchardb,
DragonBot, Excirial, Jusdafax, Estirabot, Ember of Light, BOTarate, Nblschool, Blow of Light, Koumz, SilvonenBot, WikiDao, Addbot,
Cxz111, Mortense, Fgnievinski, Jncraton, LaaknorBot, Ncsu2468, Glane23, 5 albert square, 84user, Ehrenkater, Gail, Luckas-bot, Yobot,
Amirobot, Kan8eDie, Götz, Killiondude, Jim1138, Citation bot, ArthurBot, Xqbot, Capricorn42, GrouchoBot, Mnmngb, Elemesh, A.
di M., Prari, Steve Quinn, Pinethicket, Heierjor, DLJosephson, Miracle Pen, Onel5969, NerdyScienceDude, EmausBot, Immunize,
Tommy2010, K6ka, Tolly4bolly, Thine Antique Pen, Maxrokatanski, GrayFullbuster, ClueBot NG, Anagogist, Jack Greenmaven, MelbourneStar, The Master of Mayhem, Marechal Ney, Bill bologne, Wbm1058, MusikAnimal, AdventurousSquirrel, Dbrane222, Mrt3366,
Jionpedia, BrightStarSky, Little green rosetta, Reatlas, Epicgenius, Goozkhar, Tigraan, Melody Lavender, JaconaFrere, Maadhavan Bhattacharjee, N Rajavardhan Reddy and Anonymous: 255
• Arithmetic progression Source: https://en.wikipedia.org/wiki/Arithmetic_progression?oldid=671955716 Contributors: AxelBoldt, Tarquin, William Avery, Patrick, Chas zzz brown, Michael Hardy, UserGoogol, Andres, Charles Matthews, Dcoetzee, Kbk, Hyacinth, McKay,
Fredrik, Altenmann, Nikitadanilov, Giftlite, Jackol, CryptoDerk, Tbjablin, Discospinster, Murtasa, Goochelaar, Bobo192, Robotje, C
S, .:Ajvol:., Haham hanuka, Msh210, Rh~enwiki, Snowolf, Terrible tony, Oleg Alexandrov, Hyperfusion, Sanjaymjoshi, Shreevatsa,
GregorB, Cornince, Tbone, Salvatore Ingala, Chobot, Bgwhite, Siddhant, YurikBot, Wavelength, Hairy Dude, Icedemon, KSmrq, Billon-the-Hill, Petter Strandmark, Nick, Googl, Zzuuzz, Closedmouth, RDBury, Selfworm, InverseHypercube, Melchoir, Ohnoitsjamie,
PrimeHunter, Octahedron80, DHN-bot~enwiki, Rrburke, Sebo.PL, Attys, Catapult, Aleenf1, Ckatz, 16@r, Euphrates~enwiki, MTSbot~enwiki, ST47, Goldencako, Marek69, John254, Majorly, JAnDbot, Ricardo sandoval, VoABot II, JNW, David Eppstein, Nguyễn Hữu
Dung, Ulfalizer, DarwinPeacock, Daniel5Ko, Policron, Milogardner, Jamesontai, Steel1943, VolkovBot, Johan1298~enwiki, TXiKiBoT,
Anonymous Dissident, Ravig sagi, SieBot, Rlendog, Flyer22, Masgatotkaca, Harry~enwiki, Xhackeranywhere, Denisarona, Loren.wilton,
Tai Chi Tech, ClueBot, DR23, Mild Bill Hiccup, Niceguyedc, Sohail555, DumZiBoT, XLinkBot, Gonzonoir, Kal-El-Bot, Addbot,
Macarse, Delaszk, SamatBot, Numbo3-bot, Tide rolls, Legobot, Luckas-bot, Yobot, TaBOT-zerem, Mmxx, Writer on wiki, გიგა,
AnomieBOT, Ciphers, Rubinbot, Jim1138, 9258fahsflkh917fas, Flewis, Materialscientist, Citation bot, Donanayath, Xqbot, Öncel Acar,
RibotBOT, SassoBot, LuisVillegas, Sophus Bie, A.amitkumar, Dougofborg, RTFVerterra, Robo37, Pinethicket, SpaceFlight89, RandomStringOfCharacters, Dude1818, FoxBot, DixonDBot, Duoduoduo, KurtSchwitters, Jowa fan, EmausBot, Maschen, Nikunj Pandya,
Petrb, ClueBot NG, Wcherowi, Widr, Princetct.007, Krenair, Walrus068, Kingsbrook, Sparkie82, Brad7777, Williamdemeo, Arpitkjain,
Justincheng12345-bot, Scientific Alan 2, Jochen Burghardt, Razibot, Vshender, Zorch713, Chacho39, AtoiyonTayib, Velvel2, Sahil Rally,
Arvindsingh0707, Sububu, Sourabh Tayade and Anonymous: 220
• Diophantine equation Source: https://en.wikipedia.org/wiki/Diophantine_equation?oldid=668392063 Contributors: AxelBoldt, Magnus~enwiki, XJaM, Heron, Michael Hardy, Gnomon42, Cyde, Julesd, Ruhrjung, Charles Matthews, Timwi, Dysprosia, Jitse Niesen, Robbot, Fredrik, MathMartin, Davidl9999, Robinh, Matthew Stannard, Giftlite, Zigger, Everyking, Mckaysalisbury, Bobblewik, Vivero~enwiki,
Gauss, Icairns, Dmr2, Jelammers, Chalst, Billymac00, La goutte de pluie, Haham hanuka, HasharBot~enwiki, Msh210, Burn, Hu,
Dirac1933, Oleg Alexandrov, Linas, Ajb, M412k, Chenxlee, Staecker, FlaBot, RobertG, Chobot, Siddhant, YurikBot, Wavelength, Gaius
Cornelius, Wimt, Scope creep, Reyk, Pred, Matikkapoika~enwiki, Dash77, Jsnx, SmackBot, InverseHypercube, Jagged 85, Flamarande,
Richfife, Oli Filth, Tree Biting Conspiracy, Hooriaj, Nbarth, DHN-bot~enwiki, Tsca.bot, Cícero, Wen D House, Matt Whyndham, Bidabadi~enwiki, Lambiam, Don't fear the reaper, Asyndeton, Seqsea, Az1568, Albregis, CRGreathouse, CmdrObot, Nunquam Dormio,
Myasuda, Yrodro, Fl, Sam Staton, Kazubon~enwiki, M a s, Chrislk02, Thijs!bot, LaGrange, QuiteUnusual, Magioladitis, Vanish2, David
Eppstein, Clokr, Gargiaparna, Indeed123, STBotD, DorganBot, TXiKiBoT, Hqb, Nxavar, Wtt, EnJx, Drschawrz, Yintan, ClueBot, Justin
W Smith, Plastikspork, ChandlerMapBot, J.Gowers, Nilaish, Hatsoff, Rabbit67890, Addbot, Math1353, Ronhjones, Shirtwaist, LaaknorBot, AnnaFrance, Nfogravity, Howler200, Legobot, Luckas-bot, Yobot, AnomieBOT, Onesius, Xqbot, Doulos Christos, Tobby72, Projectxanadu, BenzolBot, Kiefer.Wolfowitz, SkinnyPrude, LittleWink, Rohitphy, Logical Gentleman, Marksmith55, Archaicmath, Duoduoduo, EmausBot, KHamsun, 1curtisom, Fred Gandt, D.Lazard, Wayne Slam, Chewings72, ClueBot NG, MelbourneStar, Baseball
Watcher, Rezabot, Helpful Pixie Bot, Mokhtari34, Tekwani, GregorDS, MahdiBot, ChrisGualtieri, Deltahedron, BeaumontTaz, Jochen
Burghardt, Brirush, Rrmath28, Trompedo, Tudor987, Ghulamabbass, Arpan Mathur, MNSMUPhysicist, NyanCatGirl, Loraof, KasparBot and Anonymous: 111
• Equation Source: https://en.wikipedia.org/wiki/Equation?oldid=670209685 Contributors: AxelBoldt, Brion VIBBER, Vicki Rosenzweig, Bryan Derksen, Tarquin, Youssefsan, Christian List, Toby Bartels, Youandme, Olivier, Chas zzz brown, Michael Hardy, Dominus, Delirium, Ellywa, Iulianu, Suisui, Andres, Mxn, Pizza Puzzle, Charles Matthews, Timwi, Dysprosia, Robbot, Fredrik, Henrygb,
Alan Liefting, Giftlite, Pretzelpaws, Tom harrison, Zaphod Beeblebrox, Cap601, Karl Dickman, Abdull, CALR, Discospinster, Mani1,
Paul August, Andrejj, Bobo192, AllyUnion, Obradovic Goran, HasharBot~enwiki, Jumbuck, Orzetto, Alansohn, Cdc, Shoefly, HenryLi,
Oleg Alexandrov, Nuno Tavares, Linas, StradivariusTV, WadeSimMiser, Isnow, Zzyzx11, Mandarax, Graham87, Magister Mathematicae, BD2412, Island, Josh Parris, MarSch, Quiddity, Salix alba, FlaBot, Chobot, Nagytibi, DVdm, YurikBot, Wavelength, PiAndWhippedCream, RussBot, Chaos, Rick Norwood, Wiki alf, Caseyh, ManoaChild, Zzuuzz, Mike Dillon, Arthur Rubin, Pb30, Gesslein,
GrinBot~enwiki, Asterion, TravisTX, Sardanaphalus, Veinor, RDBury, Incnis Mrsi, KnowledgeOfSelf, Melchoir, Unyoyega, Jagged
85, Hardyplants, Gilliam, Carl.bunderson, Kurykh, Keegan, PrimeHunter, MalafayaBot, Octahedron80, Can't sleep, clown will eat me,

62

CHAPTER 8. SEQUENCE

Ioscius, Onorem, Addshore, SundarBot, Cybercobra, Jiddisch~enwiki, Lambiam, Eliyak, Heraclesprogeny, Hu12, IvanLanin, Nethac
DIU, Tawkerbot2, CRGreathouse, Scohoust, Dgw, MarsRover, Freakoclark, FilipeS, AndrewHowse, Gogo Dodo, QRX, Christian75,
Nsaum75, Thijs!bot, Epbr123, Marek69, John254, Mailseth, Seaphoto, Quintote, Danger, Karadimos, Samar, Septembrinol, Maias, Magioladitis, Bongwarrior, VoABot II, JamesBWatson, Charlielee111, Riceplaytexas, Nyttend, Cic, Bcherkas, David Eppstein, Martynas
Patasius, DerHexer, Khalid Mahmood, MartinBot, PrestonH, J.delanoy, Pharaoh of the Wizards, Numbo3, Terrek, Maurice Carbonaro,
Eliz81, Salih, McSly, Spens10, Indeed123, AntiSpamBot, NewEnglandYankee, Cometstyles, STBotD, Treisijs, BoJosley, VolkovBot,
AlnoktaBOT, TXiKiBoT, Antoni Barau, Anonymous Dissident, Ask123, Ocolon, RiverStyx23, Complex (de), Enigmaman, Synthebot,
Symane, Thebisch, Netopalis, SieBot, Ivan Štambuk, Scarian, Iamthedeus, Caltas, Happysailor, Prestonmag, Oxymoron83, Techman224,
Gordonofcartoon, Macy, Church, ClueBot, The Thing That Should Not Be, Boing! said Zebedee, Manishearth, DragonBot, Excirial,
Estirabot, Lartoven, Rejka, Jotterbot, Xxphil, Scrunter, Versus22, SoxBot III, Corz0770, BodhisattvaBot, Kal-El-Bot, PL290, Rfdhasgfjhfgvhmavjvm, Addbot, AVand, Bob is a bitch, Vchorozopoulos, CanadianLinuxUser, MrOllie, Soliquid, Kisbesbot, Sardur, Tide
rolls, Frogger3140, Aaroncrick, Qwertol, Luckas-bot, Yobot, Kan8eDie, AnomieBOT, Jim1138, Dick Beldin, Coloroftheskywatch, OllieFury, GB fan, Xqbot, Timir2, Jeffrey Mall, Grim23, NOrbeck, GrouchoBot, RibotBOT, Saalstin, FrescoBot, Xenoss, Hhhhhannah,
Gagaspocket, Ragha joshi, Pinethicket, Elockid, LittleWink, Btilm, Jujutacular, Benbeltran, GregKaye, Vrenator, Duoduoduo, Nataev,
IGraph, Danielklotz, Willnaish, Mean as custard, Alph Bot, Nlefr, EmausBot, FalseAxiom, Ibbn, TuHan-Bot, ZéroBot, Josve05a, Derekleungtszhei, Access Denied, Mjj’sbff, D.Lazard, Wayne Slam, Ocaasi, DOwenWilliams, Maschen, Airolg10, ChuispastonBot, GrayFullbuster, ClueBot NG, Jkwchui, Frietjes, Firowkp, Widr, Helpful Pixie Bot, BG19bot, Furkhaocean, Leonxlin, Akanari, Altaïr, SodaAnt,
Sparkie82, Zujua, Gelid123, ChrisGualtieri, A114112836, Jizballer245, Siuenti, Dexbot, Lugia2453, Jochen Burghardt, Makalrfekt,
Brirush, Olgakarpushin11, Mark viking, Phyzics, Jmari0818, Matty.007, Theresalwaysanorman, TROLL12345, Sarah Joy Jones, Mimo,
Ojanapothik, Gennaro Amendola 77, Muzikbox, Loraof, Asog07, Aergenteu, Ieididjdjjdid, Concaveisfag, KasparBot, This is a mobile
phone and Anonymous: 294
• Geometric progression Source: https://en.wikipedia.org/wiki/Geometric_progression?oldid=670950573 Contributors: AxelBoldt, Zundark, Tarquin, Patrick, Michael Hardy, Delirium, Conti, Charles Matthews, Dcoetzee, Hyacinth, Fredrik, Matt me, R3m0t, Mayooranathan, Henrygb, Aetheling, Tobias Bergemann, Giftlite, Knutux, LucasVB, MarkSweep, Rpchase, Jcw69, Allefant, Moxfyre, Mike
Rosoft, MuDavid, Paul August, Aranel, Touriste, Elementalish, Aisaac, Msh210, Rh~enwiki, Arthena, PAR, Mosesofmason, Justinlebar, Olethros, Gerbrant, Graham87, Yurik, Sango123, Lmatt, BradBeattie, Chobot, Sbrools, Redde, Siddhant, YurikBot, Wavelength,
Icedemon, JabberWok, Dantheox, DarthVader, Haihe, Plamka, EAderhold, Lt-wiki-bot, Arthur Rubin, Nemu, Mike1024, Pred, Hearth,
Banus, Thorney¿?, Finell, SmackBot, RDBury, Incnis Mrsi, Melchoir, Nereus124, Ixtli, Janmarthedal, Bluebot, Octahedron80, DHNbot~enwiki, Can't sleep, clown will eat me, Jratt, Mark Wolfe, Nakon, Stefano85, Vina-iwbot~enwiki, Netnubie, Jim.belk, Advance512,
Mets501, Pjrm, JForget, CmdrObot, Ichiroo, FilipeS, Haifadude, ST47, Goldencako, Tawkerbot4, Joeyfox10, Awmorp, Vanished User
jdksfajlasd, Thijs!bot, Dugwiki, AntiVandalBot, Михајло Анђелковић, JAnDbot, Leuko, Divyesikka, Gaeddal, 01001, MSBOT, JamesBWatson, Meissmart, JJ Harrison, David Eppstein, Quanticle, Andylatto, Chrisalvino, Policron, DavidCBryant, DorganBot, Gp4rts,
VolkovBot, Johan1298~enwiki, Jeff G., LokiClock, Philip Trueman, Af648, TXiKiBoT, Vertciel, SieBot, Yulu, Anchor Link Bot,
Timeastor, ClueBot, Justin W Smith, DR23, Mathwizkid, He7d3r, Fattyjwoods, NellieBly, Addbot, Laubpatr, Metagraph, Fielddaysunday, Tide rolls, Jarble, Odder, Luckas-bot, Yobot, გიგა, AnomieBOT, The Parting Glass, 9258fahsflkh917fas, Xqbot, Resident Mario,
Krishano, LuisVillegas, Joxemai, RTFVerterra, Pepper, Gas Panic42, Amgc56, Turian, FoxBot, TobeBot, Thelema418, Nascar1996,
Skittlestastegood, Jowa fan, EmausBot, Felix Hoffmann, Wikipelli, Tahdah, Slawekb, Josve05a, L Kensington, Maschen, ClueBot NG,
Wcherowi, PoqVaUSA, Ghostsarememories, Sparkie82, Rahaven, Brad7777, Anbu121, Arpitkjain, Henri.vanliempt, Amirki, 1Minow,
Brirush, Dennis at Empa Media, Jhncls, Zereth, *thing goes, Hazo11413 and Anonymous: 221
• Geometric series Source: https://en.wikipedia.org/wiki/Geometric_series?oldid=672472757 Contributors: AxelBoldt, Bryan Derksen,
The Anome, XJaM, Heron, Michael Hardy, Willsmith, Pnm, ArnoLagrange, LittleDan, Poor Yorick, Jitse Niesen, Hyacinth, Henrygb,
Per Abrahamsen, Vacuum, Giftlite, MSGJ, Vsb, Moxfyre, Rich Farmbrough, Guanabot, Paul August, Touriste, Kenyon, Mindmatrix, Salix alba, The wub, Nihiltres, Kri, Wavelength, Gillis, Closedmouth, Arthur Rubin, Reyk, Netrapt, Ghazer~enwiki, SmackBot,
Michaelliv, Incnis Mrsi, InverseHypercube, Golwengaud, Kostmo, Hgrosser, Jbergquist, Black Carrot, Jim.belk, Happy-melon, Lavateraguy, CBM, Schaber, Arrataz, Gogo Dodo, Escarbot, Uplink3r, Thenub314, JamesBWatson, JJ Harrison, David Eppstein, JaGa,
Ankitdoshi1, Eastmbr, R'n'B, Pbroks13, AstroHurricane001, Policron, Fylwind, Austinmohr, Pleasantville, LokiClock, Philip Trueman,
Ocolon, Chenzw, StevenJohnston, Oboeboy, Caltas, Yerpo, T5j6p9, Archaeogenetics, Khvalamde, Shane87, Asperal, PerryTachett, Drgarden, DonAByrd, ClueBot, Justin W Smith, DanielDeibler, Timberframe, Niceguyedc, Hans Adler, BOTarate, Eranus~enwiki, PCHSNJROTC, HiTechHiTouch, Addbot, DOI bot, Zarcadia, Jarble, Clay Juicer, Yobot, AnomieBOT, Bdmy, Dithridge, Trut-h-urts man,
Raffamaiden, NOrbeck, Hugetim, Efadae, MrHeberRomo, Citation bot 1, S iliad, Hexadecachoron, Duoduoduo, Thelema418, Bobby122,
WillNess, Ramblagir, Slawekb, 4blossoms, Souless194, VoilàY'all, DASHBotAV, Mastomer, Rocketrod1960, ClueBot NG, Wcherowi,
Helpful Pixie Bot, Jakemymath, Rahaven, Brad7777, ‫יהודה שמחה ולדמן‬, Zetazeros, OceanEngineerRI, Amirki, Webclient101, Saehry,
Stephan Kulla, Frosty, Doctordubin, Hillbillyholiday, CsDix, Gkvp, Babitaarora, ColeLoki, Bellezzasolo, Staymathy, Vrkssai, Monkbot,
Ktlabe, Tymon.r, Feitreim and Anonymous: 166
• Scale factor Source: https://en.wikipedia.org/wiki/Scale_factor?oldid=650962405 Contributors: Patrick, Boud, Ahoerstemeier, Bearcat,
Giftlite, Nichalp, Art Carlson, Antandrus, Discospinster, Bobo192, Shenme, AzaToth, Snowolf, LFaraone, P Ingerson, Gene Nygaard, Drbreznjev, Oleg Alexandrov, Linas, Jibbley, Phileas, Margosbot~enwiki, Alexjohnc3, DVdm, Gwernol, NawlinWiki, 48v, Tjarrett, Psy guy,
Jeh, Mxcatania, Nikkimaria, Pb30, Pifvyubjwm, SmackBot, Bmearns, Canthusus, Gilliam, Nbarth, Whispering, Kostmo, Onorem, Flyguy649, Akriasas, Kilonum, PseudoSudo, Hypnosifl, Mets501, Majora4, Courcelles, Woodshed, Olaf Davis, MC10, Epbr123, Daa89563,
Luna Santin, Mhaitham.shammaa, Husond, David Eppstein, MartinBot, LedgendGamer, J.delanoy, SoCalSuperEagle, Lights, Anonymous Dissident, Seraphim, HiDrNick, Caltas, Flyer22, Explicit, Beeblebrox, ClueBot, NickCT, Avenged Eightfold, Plasynins, Thingg,
Matt Millar, Versus22, Dsimic, Bhockey10, Cst17, CarsracBot, FluffyWhiteCat, 5 albert square, Ehrenkater, Tide rolls, Gail, Ben Ben,
Luckas-bot, TaBOT-zerem, II MusLiM HyBRiD II, MarcoAurelio, Daniel 1992, Ruy Pugliesi, Shirik, Doulos Christos, Smallman12q,
Pinethicket, Sirkablaam, VernoWhitney, RA0808, John Cline, ChuispastonBot, ClueBot NG, Jack Greenmaven, Wcherowi, BossMan16,
Widr, Chillllls, Qbgeekjtw, Mark Arsten, Smartyornot, Brad7777, Angelsehon999, Interlude65, Epicgenius, Eyesnore, CallmeJ, DavidLeighEllis, KarWi, Waylongh, Awesomesauce8171 and Anonymous: 216
• Sequence Source: https://en.wikipedia.org/wiki/Sequence?oldid=673701777 Contributors: AxelBoldt, Mav, Zundark, Tarquin, XJaM,
Toby Bartels, Imran, Camembert, Youandme, Lir, Patrick, Michael Hardy, Ihcoyc, Poor Yorick, Nikai, EdH, Charles Matthews, Dysprosia, Greenrd, Hyacinth, Zero0000, Sabbut, Garo, Robbot, Lowellian, MathMartin, Stewartadcock, Henrygb, Bkell, Tosha, Centrx,
Giftlite, BenFrantzDale, Lupin, Herbee, Horatio, Edcolins, Vadmium, Leonard Vertighel, Manuel Anastácio, Alexf, Fudo, Melikamp,
Sam Hocevar, Tsemii, Ross bencina, Jiy, TedPavlic, Paul August, JoeSmack, Elwikipedista~enwiki, Syp, Pjrich, Shanes, Jonathan Drain,
Nk, Obradovic Goran, Haham hanuka, Zaraki~enwiki, Merope, Jumbuck, Reubot, Jet57, Olegalexandrov, Ringbang, Djsasso, Total-

8.12. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

63

cynic, Oleg Alexandrov, Hoziron, Linas, Madmardigan53, MFH, Isnow, Graham87, Dpv, Mendaliv, Salix alba, Figs, VKokielov, Loggie, Rsenington, RexNL, Pexatus, Fresheneesz, Kri, Ryvr, Chobot, Lightsup55, Krishnavedala, Wavelength, Michael Slone, Grubber,
Arthur Rubin, JahJah, Pred, Finell, KHenriksson, Gelingvistoj, Chris the speller, Bluebot, Nbarth, Mcaruso, Suicidalhamster, SundarBot,
Dreadstar, Fagstein, Just plain Bill, Xionbox, Dreftymac, Gco, CRGreathouse, CBM, Gregbard, Cydebot, Xantharius, Epbr123, KCliffer,
Saber Cherry, Rlupsa, Marek69, Urdutext, Icep, Ste4k, Mutt Lunker, JAnDbot, Asnac, Coolhandscot, Martinkunev, VoABot II, Avjoska,
JamesBWatson, Brusegadi, Minimiscience, Stdazi, DerHexer, J.delanoy, Trusilver, Suenm~enwiki, Ncmvocalist, Belovedfreak, Policron,
JingaJenga, VolkovBot, ABF, AlnoktaBOT, Philip Trueman, Digby Tantrum, JhsBot, Isis4563, Wolfrock, Xiong Yingfei, Newbyguesses,
SieBot, Scarian, Yintan, Xelgen, Outs, Paolo.dL, OKBot, Pagen HD, Wahrmund, Classicalecon, Atif.t2, Crambo0349, ClueBot, Justin W
Smith, Fyyer, SuperHamster, Excirial, Estirabot, Jotterbot, Thingg, Downgrader, Aj00200, XLinkBot, Stickee, Rror, WikHead, Brentsmith101, Addbot, Non-dropframe, Kongr43gpen, Matěj Grabovský, Legobot, Luckas-bot, Yobot, Eric-Wester, 4th-otaku, AnomieBOT,
Jim1138, Law, Materialscientist, E2eamon, ArthurBot, Ayda D, Xqbot, Omnipaedista, RibotBOT, Charvest, Shadowjams, Thehelpfulbot, Dan6hell66, Constructive editor, Mark Renier, Tal physdancer, SixPurpleFish, Pinethicket, BRUTE, SkyMachine, PiRSquared17,
Roy McCoy, RjwilmsiBot, Tzfyr, EmausBot, John of Reading, GoingBatty, Wikipelli, K6ka, Brent Perreault, Nellandmice, Bethnim,
Ida Shaw, Alpha Quadrant, KuduIO, D.Lazard, SporkBot, Wayne Slam, Donner60, Chewings72, ClueBot NG, Satellizer, Widr, MerlIwBot, Helpful Pixie Bot, HMSSolent, Curb Chain, Calabe1992, Brad7777, Minsbot, Praxiphenes, EuroCarGT, Ven Seyranyan., Jegyao,
DavyRalph, Graphium, Jochen Burghardt, Brirush, Mark viking, LoMaPh, Immonster, EricsonWillians, Emlynlee, Buscus 3, JackHoang,
BemusedObserver, Some1Redirects4You and Anonymous: 210

8.12.2

Images

• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public domain Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)

• File:Cartesian-coordinate-system-with-circle.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2e/Cartesian-coordinate-system-with-circle
svg License: CC-BY-SA-3.0 Contributors: ? Original artist: User 345Kai on en.wikipedia
• File:Cauchy_sequence_illustration.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/62/Cauchy_sequence_illustration.
svg License: CC0 Contributors: Own work Original artist: Krishnavedala
• File:Cauchy_sequence_illustration2.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7a/Cauchy_sequence_illustration2.
svg License: Public domain Contributors: Based on File:Cauchy_sequence_illustration2.png by Oleg Alexandrov Original artist: Own work
• File:Coniques_cone.png Source: https://upload.wikimedia.org/wikipedia/commons/5/51/Coniques_cone.png License: CC-BY-SA-3.0
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• File:Converging_Sequence_example.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e4/Converging_Sequence_example.
svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Melikamp
• File:E-to-the-i-pi.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/E-to-the-i-pi.svg License: CC BY 2.5 Contributors: ? Original artist: ?
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• File:Fibonacci_blocks.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/bc/Fibonacci_blocks.svg License: Public domain Contributors: Own work Original artist: ElectroKid (☮ • ✍)
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• File:Geometric_Segment.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/ab/Geometric_Segment.svg License: Public
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• File:Lorentz.PNG Source: https://upload.wikimedia.org/wikipedia/commons/3/3c/Lorentz.PNG License: Public domain Contributors:
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• File:Merge-arrows.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/52/Merge-arrows.svg License: Public domain Contributors: ? Original artist: ?

64

CHAPTER 8. SEQUENCE

• File:Nuvola_apps_edu_mathematics_blue-p.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_
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Contributors:
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Tkgd2007
• File:Rtriangle.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6f/Rtriangle.svg License: CC-BY-SA-3.0 Contributors:
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domain Contributors: Vector version of Image:Wiktionary-logo-en.png. Original artist: Vectorized by Fvasconcellos (talk · contribs),
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• File:
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8.12.3

Content license

• Creative Commons Attribution-Share Alike 3.0