Series using mathematical induction

A series $\{a_n\}$ has an initial starting value of $a_1 =2$ . For all integers $n$ , the two roots $\alpha$ and $\beta$ of the equation $a_n x^2 - a_{n+1} x+1=0$ satisfy the following condition: $\alpha - 2 \alpha \beta + \beta = 3.$ What is the minimum positive integer $n$ that satisfies $a_n>1000?$