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

a1, a2, ..., ap,  a1, a2, ..., ap,  a1, a2, ..., ap, ...

The number p of repeated terms is called the period (period).[1]



A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a1, a2, a3, ... satisfying

an+p = an

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.



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 : XX 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.



Partial Sums

  Where k and m<p are natural numbers.

Partial Products

  Where k and m<p are natural numbers.

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:


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.



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 x1x2x3, ... is asymptotically periodic if there exists a periodic sequence a1a2a3, ... 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, ....


  1. ^ a b c d "Ultimately periodic sequence", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  2. ^ Bosma, Wieb. "Complexity of Periodic Sequences" (PDF). Retrieved 13 August 2021.
  3. ^ 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.
  4. ^ Hosch, William L. (1 June 2018). "Rational number". Encyclopedia Britannica. Retrieved 13 August 2021.