Editing Periodic sequence (section)

==Examples==
Every constant function is 1-periodic.

The sequence <math>1,2,1,2,1,2\dots</math> is periodic with least period 2.

The sequence of digits in the [[decimal]] expansion of 1/7 is periodic with period 6:

:<math>\frac{1}{7} = 0.142857\,142857\,142857\,\ldots</math>

More generally, the sequence of digits in the decimal expansion of any [[rational number]] is eventually periodic (see below).<ref>{{Cite web|last=Hosch|first=William L.|date=1 June 2018|title=Rational number|url=https://www.britannica.com/science/rational-number|access-date=13 August 2021|website=Encyclopedia Britannica|language=en}}</ref>

The sequence of powers of &minus;1 is periodic with period two:

:<math>-1,1,-1,1,-1,1,\ldots</math>

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 (group theory)|order]] in a [[group (mathematics)|group]].

A [[periodic point]] for a function {{math|''f'' : ''X'' → ''X''}} is a point {{mvar|x}} whose [[orbit (dynamics)|orbit]]

:<math>x,\, f(x),\, f(f(x)),\, f^3(x),\, f^4(x),\, \ldots</math>

is a periodic sequence.  Here, <math>f^n(x)</math> means the {{nowrap|{{mvar|n}}-fold}} [[Function composition|composition]] of {{mvar|f}} applied to {{mvar|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.