{{short description|Sequence for which the same terms are repeated over and over}} {{CS1 config|mode=cs1}} In mathematics, a '''periodic sequence''' (sometimes called a '''cycle''' or '''orbit''') is a sequence for which the same terms are repeated over and over:
:''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''p''</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''p''</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''p''</sub>, ...
The number ''p'' of repeated terms is called the '''period''' (period).<ref name=":0">{{eom|title=Ultimately periodic sequence}} </ref>
==Definition== A '''(purely) periodic''' sequence (with '''period ''p'''''), or a '''''p-''periodic sequence''', is a sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... satisfying
:''a''<sub>''n''+''p''</sub> = ''a''<sub>''n''</sub> for all values of ''n''.<ref name=":0" /><ref>{{Cite web|last=Bosma|first=Wieb|title=Complexity of Periodic Sequences|url=https://www.math.ru.nl/~bosma/pubs/periodic.pdf|access-date=13 August 2021|website=www.math.ru.nl}}</ref><ref name=":2">{{Cite journal|last1=Janglajew|first1=Klara|last2=Schmeidel|first2=Ewa|date=2012-11-14|title=Periodicity of solutions of nonhomogeneous linear difference equations|journal=Advances in Difference Equations|volume=2012|issue=1|pages=195|doi=10.1186/1687-1847-2012-195|s2cid=122892501|issn=1687-1847|doi-access=free}}</ref> 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.<ref name=ccd>{{cite book | last1 = Beck | first1 = Matthias | last2 = Robins | first2 = Sinai | contribution = Chapter 7: Finite Fourier analysis | doi = 10.1007/978-0-387-46112-0_7 | isbn = 9780387291390 | location = New York | pages = 123–137 | publisher = Springer | series = Undergraduate Texts in Mathematics | title = Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra}}</ref> The smallest ''p'' for which a periodic sequence is ''p''-periodic is called its '''least period'''<ref name=":0" /> or '''exact period'''.
==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 −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 in a group. Every periodic sequence of numbers can be written as a polynomial <math>p(x)</math>, evaluated at the powers of a root of unity: <math>a_i=p(z^i)</math> where <math>z</math> is a root of unity whose order is the period of the sequence.<ref name=ccd/>
A periodic point for a function {{math|''f'' : ''X'' → ''X''}} is a point {{mvar|x}} whose 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}} 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.
==Partial sums and products == :<math>\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}</math>,
where <math>m < p</math> and <math>k</math> 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:
:<math>\sum_{k=0}^0 \cos \left(2\pi\frac{nk}{1}\right)/1 = 1,1,1,1,1,1,1,1,1, \cdots</math>
:<math>\sum_{k=0}^{1} \cos \left(2\pi\frac{nk}{2}\right)/2 = 1,0,1,0,1,0,1,0,1,0, \cdots</math>
:<math>\sum_{k=0}^{2} \cos \left(2\pi\frac{nk}{3}\right)/3 = 1, 0,0,1,0,0,1,0,0,1,0,0,1,0,0, \cdots</math>
:<math>\cdots</math>
:<math>\sum_{k=0}^{N-1} \cos \left(2\pi\frac{nk}{N}\right)/N = 1,0,0,0,\cdots,1, \cdots \quad \text{sequence with period } N </math>
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.
==Generalizations== A sequence is '''eventually periodic''' or '''ultimately periodic'''<ref name=":0" /> if it can be made periodic by dropping some finite number of terms from the beginning. Equivalently, the last condition can be stated as <math>a_{k+r} = a_k</math> 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 ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ... is asymptotically periodic if there exists a periodic sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... for which
:<math>\lim_{n\rightarrow\infty} x_n - a_n = 0.</math><ref name=":2" />
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, ....
== References == {{Reflist}}{{Series (mathematics)}}
{{DEFAULTSORT:Periodic Sequence}} Category:Sequences and series