Periodic sequences | Mathematico

A sequence is periodic if it keeps repeating the same cycle over and over, for example

1, 4, 5, 1, 4, 5, 1, 4, 5, \dots

The period of this sequence is $3,$ because that is the length of the cycle.

For this sequence, we have

\begin{aligned} u_{4} & = u_{1} \\ u_{5} & = u_{2} \\ u_{6} & = u_{3} \\ ⋮ \end{aligned}

In general

u_{n + 3} = u_{n}

For any periodic sequence of period $k,$ we have

u_{n + k} = u_{n}

The periodic sequence $u_{n}$ begins

2, 1, \frac{7}{2}, 7, \dots

Given that the period of the sequence is $44$ :