# Recurring Styles #12 - Limit of a Recurrence!

Calculus Level 5

$\large{ S = \sum_{n=1}^\infty \dfrac{a_{2n+2}}{a^2_{n-1} a^2_{n+1}} }$

Let $$(a_n)$$ be a sequence defined by $$a_0 = 1, a_1 = 2$$, and for $$n \geq 2$$, $$a_n = a_{n-1} + a_{n-2}$$. If $$S$$ can be expressed as $$\dfrac{A}{B}$$ where $$A,B \in \mathbb Z^+, \ \gcd(A,B)=1$$. Submit the value of $$A+B$$ as your answer.

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