Series using mathematical induction

Algebra Level pending

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 \(a_n>1000?\)

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