ÖMO 1998 Problem 3

Let $a_n$ be a sequence recursively defined by $a_0 = 0, a_1 = 1$ and $a_{n+2} = a_{n+1} + a_n$ . Calculate the sum of $a_n\left( \frac 25\right)^n$ for all positive integers $n$ . For what value of the base $b$ we get the sum $1$ ?