Question 99
Let $\langle f_n \rangle$ be the Fibonacci sequence, that is, $f_0 = 0$, $f_1 = 1$, and that $f_{n+2} = f_{n+1} + f_n$ holds for all nonnegative integers $n$.
Find all pairs $(a, b)$ of positive integers with $a \lt b$ such that $f_n - 2n \cdot a^n$ is divisible by $b$ for all positive integers $n$.
Find all pairs $(a, b)$ of positive integers with $a \lt b$ such that $f_n - 2n \cdot a^n$ is divisible by $b$ for all positive integers $n$.