I think approximation can be the best way to answer these types of multiple choice questions. Since its a quadratic, you know alpha is about 6 and beta is abou

I think approximation can be the best way to answer these types of multiple choice questions. Since its a quadratic, you know alpha is about 6 and beta is about -0.3. Then, since the answer is a whole number, and we're raising to high powers, you throw away the -0.3 and approximate a_n = 6^n. Divide through by a_8 and you get (6^2 - 2) / 12, which is very close to 3.