In the infinite sequence $A$, $A_n=x^{n-1}+x^n+x^{n+1}+x^{n+2}+x^{n+3}$ where $x$ is a positive integer constant. For what value of $n$ is the ratio of $A_n$ to $x(1+x(1+x(1+x(1+x))))$ equal to $x^5$?

This one took me a minute. The trick is realizing that x(1+x(1+x(1+x(1+x)))) is actually a geometric-looking expansion. I factored out and it matched a finite sum. Clever question.

Is the denominator equal to x+x^2+x^3+x^4+x^5? If so then A_n/(that) = x^5 means A_n = x^5 * (x+x^2+x^3+x^4+x^5). Then A_n would have 5 terms starting from x^6? So n-1=6 -> n=7?

655-705 my foot, this felt harder than the rating. The nested parentheses messed with my head until I realized it's a geometric series in disguise.

Thanks, the factorization of the denominator is the key insight. Once you see it as x(1+x+x^2+x^3+x^4) it becomes straightforward.