Question 9
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 $n \ge 1,$ the terms $a_{2n-1},$ $a_{2n},$ and $a_{2n+1}$ 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 $n + a_n.$