Lately, I need to prove a random walk on n-cycle is periodic when n is even. The definition of random walk is
This means I need to show gcd of $T(x)$ is 2 for every $x \in \mathbb{Z}_n$.
First, I can show that $P^2(x,x) = 2$ for every $x$.
Then, suppose for contradiction that there is some odd $t$ such that $P^t(x,x) > 0$. Since $t$ is odd, we can write $t=2k+1$, for some $k$. Since $x$ return to $x$ after $2k+1$ steps, we have
$$x \equiv x + (2k+1) \mod n.$$
Thus, we can imply $n$ divides $2k+1$, which gives a contradiction since $n$ is even.
Is it true? I'm not sure about $x \equiv x + (2k+1) \mod n$. May someone please check it for me?
Any advice is highly appreciated.