# A Typical Recurrence Relation

Calculus Level 4

Consider the sum, $S = \sum _{n = 0} ^{\infty} \frac{a_n}{5^{2n}}$

where $$a_n$$ is a sequence defined by recurrence relation

$a_{n+2} = 2a_{n+1} + a_n \quad \forall \, n \in \mathbb Z^*$

and $$a_0 = a_1 = 1$$. If $$S = \dfrac{p}{q}$$, then find $$p+q$$.

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