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In mathematics, a **periodic sequence** 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**.

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

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

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

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

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

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:

- −1, +1, −1, +1, −1, +1, ...

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 ƒ: *X* → *X* is a point *p* whose orbit

is a periodic sequence. Periodic points are important in the theory of dynamical systems.

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:

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

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, ....