Question 11
Let $M_n$ be the $n \times n$ matrix with entries as follows: for $1 \le i \le n,$ $m_{i,i} = 10;$ for $1 \le i \le n - 1,$ $m_{i+1,i} = m_{i,i+1} = 3;$ all other entries in $M_n$ are zero. Let $D_n$ be the determinant of 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 $p + q.$
Note: The determinant of the $1 \times 1$ matrix $[a]$ is $a,$ and the determinant of the $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $ad - bc;$ for $n \ge 2,$ the determinant of an $n \times n$ matrix with first row or first column $a_1$ $a_2$ $a_3$ $\ldots$ $a_n$ is equal to $a_1C_1 - a_2C_2$ $+ a_3C_3 - \cdots$ $+ (-1)^{n+1}a_nC_n,$ where $C_i$ is the determinant of the $(n - 1) \times (n - 1)$ matrix formed by eliminating the row and column containing $a_i.$
Note: The determinant of the $1 \times 1$ matrix $[a]$ is $a,$ and the determinant of the $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $ad - bc;$ for $n \ge 2,$ the determinant of an $n \times n$ matrix with first row or first column $a_1$ $a_2$ $a_3$ $\ldots$ $a_n$ is equal to $a_1C_1 - a_2C_2$ $+ a_3C_3 - \cdots$ $+ (-1)^{n+1}a_nC_n,$ where $C_i$ is the determinant of the $(n - 1) \times (n - 1)$ matrix formed by eliminating the row and column containing $a_i.$