2011 AIME II Problems/Problem 11

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Problem:

Let $M_{n}$ be the n x n matrix with entries as follows: for $1 \leq i \leq n$, $m_{i,i} = 10$; for $1 \leq i \leq n-1$, $m_{i,i+1} = m_{i+1,i} = 3$; all other entries in $M_{n}$ are zero. Let $D_{n}$ be the determinant of the matrix $M_{n}$. Then $\sum_{n = 1}^{\infty}\frac{1}{8D_{n} + 1}$ can be represented as $\frac{p}{q}$ where p and q are relatively prime positive integers. Find p + q.

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