## Spring 2017, problem 47

Given a sequence $a_1,a_2,a_3,\ldots$ of positive integers in which every positive integer occurs exactly once, show that there exist integers $\ell$ and $m,\ 1\lt\ell \lt m$, such that $a_1+a_m=2a_{\ell}$.

Let $\ell$ be the smallest index for which $a_\ell > a_1$. Then choose $m > \ell$ so that $a_m = 2a_\ell - a_1.$ Such an $m$ exists since $2a_\ell - a_1$ occurs somewhere in the sequence, and $2a_\ell - a_1 > a_\ell,$ so it can't occur before $\ell$.