In mathematics, a periodic sequence (sometimes called a cycle) 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). Definition 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. 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 or exact period. Examples 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). 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 is a point whose orbit is a periodic sequence. Here, means the composition of applied to . 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 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: Generalizations 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 ultimately periodic if it satisfies the condition for some r and