Question 149
Let $n$ be a positive integer and let $S$ be the set $\{1,2, \ldots, n\}$. Define a function $f: S \rightarrow S$ by
$$
f(x)= \begin{cases}2 x & \text{ if } 2 x \le n, \\ 2 n-2 x+1 & \text{ otherwise. }\end{cases}
$$
Define $f^{2}(x)=f(f(x)), f^{3}(x)=f(f(f(x)))$, and so on. If $m$ is a positive integer satisfying $f^{m}(1)=1$, prove that $f^{m}(k)=k$ for all $k \in S$.
$$
f(x)= \begin{cases}2 x & \text{ if } 2 x \le n, \\ 2 n-2 x+1 & \text{ otherwise. }\end{cases}
$$
Define $f^{2}(x)=f(f(x)), f^{3}(x)=f(f(f(x)))$, and so on. If $m$ is a positive integer satisfying $f^{m}(1)=1$, prove that $f^{m}(k)=k$ for all $k \in S$.