# Periodic sequence

__: Sequence for which the same terms are repeated over and over__

**Short description**

In mathematics, a **periodic sequence** (sometimes called a **cycle**) is a sequence for which the same terms are repeated over and over:

*a*_{1},*a*_{2}, ...,*a*_{p},*a*_{1},*a*_{2}, ...,*a*_{p},*a*_{1},*a*_{2}, ...,*a*_{p}, ...

The number *p* of repeated terms is called the **period** (period).

## Definition

A periodic sequence is a sequence *a*_{1}, *a*_{2}, *a*_{3}, ... satisfying

*a*_{n+p}=*a*_{n}

for all values of *n*. If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.

## Examples

The sequence of digits in the decimal expansion of 1/7 is periodic with period 6:

- [math]\displaystyle{ \frac{1}{7} = 0.142857\,142857\,142857\,\ldots }[/math]

More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).

The sequence of powers of −1 is periodic with period two:

- [math]\displaystyle{ -1,1,-1,1,-1,1,\ldots }[/math]

More generally, the sequence of powers of any root of unity is periodic. The same holds true for the powers of any element of finite order in a group.

A periodic point for a function *f* : *X* → *X* is a point x whose orbit

- [math]\displaystyle{ x,\, f(x),\, f(f(x)),\, f^3(x),\, f^4(x),\, \ldots }[/math]

is a periodic sequence. Here, [math]\displaystyle{ f^n(x) }[/math] means the n-fold composition of f applied to x. Periodic points are important in the theory of dynamical systems. Every function from a finite set to itself has a periodic point; cycle detection is the algorithmic problem of finding such a point.

## Periodic 0, 1 sequences

Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones. Periodic zero and one sequences can be expressed as sums of trigonometric functions:

- [math]\displaystyle{ \sum_{k=1}^{1} \cos (-\pi\frac{n(k-1)}{1})/1 = 1,1,1,1,1,1,1,1,1... }[/math]

- [math]\displaystyle{ \sum_{k=1}^{2} \cos (2\pi\frac{n(k-1)}{2})/2 = 0,1,0,1,0,1,0,1,0... }[/math]

- [math]\displaystyle{ \sum_{k=1}^{3} \cos (2\pi\frac{n(k-1)}{3})/3 = 0,0,1,0,0,1,0,0,1,0,0,1,0,0,1... }[/math]

- [math]\displaystyle{ ... }[/math]

- [math]\displaystyle{ \sum_{k=1}^{N} \cos (2\pi\frac{n(k-1)}{N})/N = 0,0,0...,1 \text{ sequence with period } N }[/math]

## Generalizations

A sequence is **eventually periodic** if it can be made periodic by dropping some finite number of terms from the beginning. For example, the sequence of digits in the decimal expansion of 1/56 is eventually periodic:

- 1 / 56 = 0 . 0 1 7 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 ...

A sequence is **asymptotically periodic** if its terms approach those of a periodic sequence. That is, the sequence *x*_{1}, *x*_{2}, *x*_{3}, ... is asymptotically periodic if there exists a periodic sequence *a*_{1}, *a*_{2}, *a*_{3}, ... for which

- [math]\displaystyle{ \lim_{n\rightarrow\infty} x_n - a_n = 0. }[/math]

For example, the sequence

- 1 / 3, 2 / 3, 1 / 4, 3 / 4, 1 / 5, 4 / 5, ...

is asymptotically periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, ....

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Original source: https://en.wikipedia.org/wiki/Periodic sequence.
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