2011 AIME II Problems/Problem 11

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.