In mathematics, a periodic sequence (sometimes called a cycle or orbit) is a sequence for which the same terms are repeated over and over: The number p of repeated terms is called the period (period).
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).
The same holds true for the powers of any element of finite order in a group.
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.
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.