lean_workbook_plus_7511

Let $\alpha, \beta$ be the roots of the equation $x^2 - x - 1 = 0$, and let $a_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}, n = 1, 2, \ldots$.\na) Prove that for any positive integer $n$, $a_{n+2} = a_{n+1} +a_n$\nb) Find all positive integers $a, b$ $(a < b)$ such that for any positive integer $n$, $b$ divides $a_n - 2na^n$.