# Periodic sequence

In mathematics, a **periodic sequence** (sometimes called a **cycle** or **orbit**) 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).^{[1]}

## Definition

[edit]A **(purely) periodic** sequence (with **period p**), or a

**, is a sequence**

*p-*periodic sequence*a*

_{1},

*a*

_{2},

*a*

_{3}, ... satisfying

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

for all values of *n*.^{[1]}^{[2]}^{[3]} 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.^{[citation needed]} The smallest *p* for which a periodic sequence is *p*-periodic is called its **least period**^{[1]} or **exact period**.

## Examples

[edit]Every constant function is 1-periodic.

The sequence is periodic with least period 2.

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

More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below).^{[4]}

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

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

is a periodic sequence. Here, 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.

## Identities

[edit]### Partial Sums

[edit]- Where k and m<p are natural numbers.

### Partial Products

[edit]- Where k and m<p are natural numbers.

## Periodic 0, 1 sequences

[edit]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:

One standard approach for proving these identities is to apply De Moivre's formula to the corresponding root of unity. Such sequences are foundational in the study of number theory.

## Generalizations

[edit]A sequence is **eventually periodic** or **ultimately periodic**^{[1]} if it can be made periodic by dropping some finite number of terms from the beginning. Equivalently, the last condition can be stated as for some *r* and sufficiently large *k*. 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

^{[3]}

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, ....

## References

[edit]- ^
^{a}^{b}^{c}^{d}"Ultimately periodic sequence",*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] **^**Bosma, Wieb. "Complexity of Periodic Sequences" (PDF).*www.math.ru.nl*. Retrieved 13 August 2021.- ^
^{a}^{b}Janglajew, Klara; Schmeidel, Ewa (2012-11-14). "Periodicity of solutions of nonhomogeneous linear difference equations".*Advances in Difference Equations*.**2012**(1): 195. doi:10.1186/1687-1847-2012-195. ISSN 1687-1847. S2CID 122892501. **^**Hosch, William L. (1 June 2018). "Rational number".*Encyclopedia Britannica*. Retrieved 13 August 2021.