A sequence is called periodic if it repeats itself over and over again at regular intervals.
Formally, a sequence \(u_1\), \(u_2\), … is periodic with period \(T\) (where \(T>0\)) if \(u_{n+T}=u_n\) for all \(n\ge 1\). The smallest such \(T\) is called the least period (or often just “the period”) of the sequence.
For example, the following sequences are periodic:
\[\begin{align*} &0,\ 1,\ 0,\ 1,\ 0,\ 1,\ \dotsc\ &&\text{least period $2$}\\ &0,\ 1,\ 0,\ {-1},\ 0,\ 1,\ 0,\ {-1},\ \dotsc\ &&\text{least period $4$}\\ &1,\ 1,\ 1,\ 1,\ 1,\ \dotsc\ &&\text{least period $1$} \end{align*}\]
