Question

A sequence of positive integers with $a_1 = 1$ and $a_9 + a_{10} = 646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\ge1$, the terms $a_{2n - 1}$, $a_{2n}$, $a_{2n + 1}$ are in geometric progression, and the terms $a_{2n}$, $a_{2n + 1}$, and $a_{2n + 2}$ are in arithmetic progression. Let $a_n$ be the greatest term in this sequence that is less than 1000. Find $a_n$.