Question 81
Let $a_1, a_2, \dots$ be a sequence of positive integers such that
$$
a_{n+1} = \begin{cases} \frac{a_n}{2} & \text{if } a_n \text{ is even,} \\ 2^r + \frac{a_n+1}{2} & \text{if } a_n \text{ is odd and } 2^{r-1} \le a_n \lt 2^r. \end{cases}
$$
Prove that no matter what the value of $a_1$ is, there exists an $N$ such that for all $n \gt N$, $a_n = a_{n+2}$.
$$
a_{n+1} = \begin{cases} \frac{a_n}{2} & \text{if } a_n \text{ is even,} \\ 2^r + \frac{a_n+1}{2} & \text{if } a_n \text{ is odd and } 2^{r-1} \le a_n \lt 2^r. \end{cases}
$$
Prove that no matter what the value of $a_1$ is, there exists an $N$ such that for all $n \gt N$, $a_n = a_{n+2}$.