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<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub>, a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub>, a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>p</sub>, ...
The number p of repeated terms is called the period (period).
Definition
A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, ... satisfying
:a<sub>n+p</sub> = a<sub>n</sub>
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. The smallest p for which a periodic sequence is p-periodic is called its least period
The sequence of powers of −1 is periodic with period two:
:<math>-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. Every periodic sequence of numbers can be written as a polynomial <math>p(x)</math>, evaluated at the powers of a root of unity: <math>a_i=p(z^i)</math> where <math>z</math> is a root of unity whose order is the period of the sequence.