---
tag: Q3B6K9HYRG
kind: concept
name: Periodic sequence
msc: [39A23, 11B83]
sources:
  - url: https://en.wikipedia.org/wiki/Periodic_sequence
same_as:
  wikidata: Q3503302
  wikipedia: Periodic sequence
---

## Summary

A periodic sequence is a [[sequence|0G9EKM0E15]] that repeats the same finite block of terms forever.
The length of the shortest repeating block is its least period.
Periodic sequences are the discrete analogue of [[periodic functions|0ATF6Q0633]].
They appear as periodic solutions of difference equations, as decimal expansions of rational numbers, and as Fibonacci numbers modulo a fixed integer.

## Explanations

### L1 human

A periodic sequence is a pattern that repeats like a necklace with a fixed set of beads.
The sequence $1, 2, 3, 1, 2, 3, \dots$ repeats after $3$ terms, so it has period $3$.
The digits after the decimal point of $1/7 = 0.142857142857\ldots$ form a periodic sequence of period $6$.
A constant sequence is periodic with period $1$.