Question 24
In a sequence $u_{0}, u_{1}, \ldots$ of positive integers, $u_{0}$ is arbitrary, and for any non-negative integer $n$,
$$
u_{n+1}= \begin{cases}\frac{1}{2} u_{n} & \text{ for even } u_{n} \\ a+u_{n} & \text{ for odd } u_{n}\end{cases}$$
where $a$ is a fixed odd positive integer. Prove that the sequence is periodic from a certain step.
$$
u_{n+1}= \begin{cases}\frac{1}{2} u_{n} & \text{ for even } u_{n} \\ a+u_{n} & \text{ for odd } u_{n}\end{cases}$$
where $a$ is a fixed odd positive integer. Prove that the sequence is periodic from a certain step.