A [sequence](https://artofproblemsolving.com/wiki/index.php/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](https://artofproblemsolving.com/wiki/index.php/Geometric_sequence), the second, third, and fourth terms are in [arithmetic progression](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence), 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 $n+a_n.$
Let $x = a_2$; then solving for the next several terms, we find that $a_3 = x^2,\ a_4 = x(2x-1),\ a_5$ $= (2x-1)^2,\ a_6$ $= (2x-1)(3x-2)$, and in general, $a_{2n} = f(n-1)f(n)$, $a_{2n+1} = f(n)^2$, where $f(n) = nx - (n-1)$.[[1](https://artofproblemsolving.com#note_1)]
From \[a_9 + a_{10} = f(4)^2 + f(4)f(5) = (4x-3)(9x-7) = 646 = 2\cdot 17 \cdot 19\], we find that by either the [quadratic formula](https://artofproblemsolving.com/wiki/index.php/Quadratic_formula) or trial-and-error/modular arithmetic that $x=5$. Thus $f(n) = 4n+1$, and we need to find the largest $n$ such that either $f(n)^2\, \mathrm{or}\, f(n)f(n-1) < 1000$. This happens with $f(7)f(8) = 29 \cdot 33 = 957$, and this is the $2(8) = 16$th term of the sequence.
The answer is $957 + 16 = \boxed{973}$.

[^](https://artofproblemsolving.com#ref_1) We can show this by simultaneous [induction](https://artofproblemsolving.com/wiki/index.php/Induction): since 
\begin{align*}a_{2n} &= 2a_{2n-1} - a_{2n-2} = 2a_{2(n-1)+1} - a_{2(n-1)} \\ &= 2f(n-1)^2 - f(n-2)f(n-1) \\ &= f(n-1)[2f(n-1) - f(n-2)] \\ &= f(n-1)[(2n-2-n+2)x-(2n-4-n+3)] \\ &= f(n-1)f(n) \end{align*}
and
\begin{align*}a_{2n+1} &= \frac{a_{2n}^2}{a_{2n-1}} = \frac{f(n-1)^2f(n)^2}{f(n-1)^2} = f(n)^2 \\ \end{align*}