Periodic sequence

A periodic sequence is a term used in mathematics . In this specific class of sequences , the following terms repeat themselves after a specific period length.

definition

A sequence is called periodic if there are natural numbers and such that ${\displaystyle a_{1},a_{2},a_{3},\ldots }$ ${\displaystyle d}$ ${\displaystyle N}$

${\displaystyle a_{n}=a_{n+d}}$

applies to all . In the case , the sequence is called purely periodic or strictly periodic. The minimum number with the above property is called the period length. ${\displaystyle n>N}$ ${\displaystyle N=0}$ ${\displaystyle d}$

example

Let and for , where is the modulo operator. ${\displaystyle a_{0}=1}$ ${\displaystyle a_{n+1}=2a_{n}{\bmod {1}}00}$ ${\displaystyle n\geq 0}$ ${\displaystyle {}{\bmod {}}}$

The number formed from the last two digits of the decimal representation of is clear . This sequence begins with the values ${\displaystyle a_{n}}$ ${\displaystyle 2^{n}}$

1, 2, 4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 4 ...

Then these values ​​are repeated.

If one considers a recursively defined sequence in general, i.e. a sequence that is defined by for a fixed function , and only assumes finitely many values, then the sequence is always periodic. ${\displaystyle a_{n+1}=f(a_{n})}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle a_{i}}$

Periodic digit sequences

Let it be a fixed natural number. If and are natural numbers, then the sequence of decimal places of the - adic representation of becomes periodic according to the above principle, because it is determined iteratively by the remainders in the division , and these remainders can only assume the finitely many values . ${\displaystyle b>1}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle b}$ ${\displaystyle {\tfrac {p}{q}}}$ ${\displaystyle 0,1,\ldots ,b-1}$

Individual evidence

1. ↑ Uwe Storch, Hartmut Wiebe: Textbook of Mathematics, Volume II: Linear Algebra . BI-Wissenschafts-Verlag, 1990, ISBN 3-411-14101-8 , p. 305.
2. ↑ ^{a b} Periodic sequence . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .