USAMO 1994 Problem 1

$a_{1}, a_{2}, a_{3}, \ldots$ are positive integers such that $a_{n}>a_{n-1}+1$. Put $b_{n}=a_{1}+a_{2}+\ldots+a_{n}$. Show that there is always a square in the range $b_{n}, b_{n}+1, b_{n}+2, \ldots, b_{n+1}-1$.