CWMO 2002 Problem 7

Assume that $\alpha$ and $\beta$ are two roots of the equation: $x^2-x-1=0$ . Let $a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ , $n=1, 2, \cdots$ . (1) Prove that for any positive integer $n$ , we have $a_{n+2}=a_{n+1}+a_n$ . (2) Find all positive integers $a$ and $b$ , $a<b$ , satisfying $b \mid a_n-2na^n$ for any positive integer $n$ .