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).
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 1 , 2 , 1 , 2 , 1 , 2 … {\displaystyle 1,2,1,2,1,2\dots } is periodic with least period 2. The sequence of digits in the decimal expansion of 1/7 is periodic with period 6:
1 7 = 0.142857 142857 142857 … {\displaystyle {\frac {1}{7}}=0.142857\,142857\,142857\,\ldots }
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 , … {\displaystyle -1,1,-1,1,-1,1,\ldots }
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 p ( x ) {\displaystyle p(x)} , evaluated at the powers of a root of unity: a i = p ( z i ) {\displaystyle a_{i}=p(z^{i})} where z {\displaystyle z} is a root of unity whose order is the period of the sequence. A periodic point for a function f : X → X is a point x whose orbit
x , f ( x ) , f ( f ( x ) ) , f 3 ( x ) , f 4 ( x ) , … {\displaystyle x,\,f(x),\,f(f(x)),\,f^{3}(x),\,f^{4}(x),\,\ldots }
is a periodic sequence. Here, f n ( x ) {\displaystyle f^{n}(x)} 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 and products
∑ n = 1 k p + m a n = k ∗ ∑ n = 1 p a n + ∑ n = 1 m a n , ∏ n = 1 k p + m a n = ( ∏ n = 1 p a n ) k ⋅ ∏ n = 1 m a n {\displaystyle \sum _{n=1}^{kp+m}a_{n}=k*\sum _{n=1}^{p}a_{n}+\sum _{n=1}^{m}a_{n},\qquad \prod _{n=1}^{kp+m}a_{n}={\biggl (}{\prod _{n=1}^{p}a_{n}}{\biggr )}^{k}\cdot \prod _{n=1}^{m}a_{n}} , where m < p {\displaystyle m<p} and k {\displaystyle k} are positive integers.
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:
∑ k = 0 0 cos ( 2 π n k 1 ) / 1 = 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , ⋯ {\displaystyle \sum _{k=0}^{0}\cos \left(2\pi {\frac {nk}{1}}\right)/1=1,1,1,1,1,1,1,1,1,\cdots }
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