Problem

Source: CWMO 2002 P7

Tags: induction, number theory proposed, number theory



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$.