Question 18
Let $r$ be a positive integer, and let $a_{0}, a_{1}, \ldots$ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
$$
a_{m}+a_{m+1}+\cdots+a_{m+s}=a_{n}+a_{n+1}+\cdots+a_{n+s}
$$
Prove that the sequence is periodic, i.e. there exists some $p \geqslant 1$ such that $a_{n+p}=a_{n}$ for all $n \geqslant 0$.
$$
a_{m}+a_{m+1}+\cdots+a_{m+s}=a_{n}+a_{n+1}+\cdots+a_{n+s}
$$
Prove that the sequence is periodic, i.e. there exists some $p \geqslant 1$ such that $a_{n+p}=a_{n}$ for all $n \geqslant 0$.