Problem Analysis #43

Let $\{a_n\}$ be an arithmetic sequence and $\{b_n\}$ a geometric sequence whose first terms and common difference/ratio are equal. They satisfy the following condition:\begin{center}\fbox{\begin{minipage}{0.9\textwidth}For some natural number $k$,$b_{k+i} = \frac{1}{a_i} - 1 \quad (i = 1, 2, 3).$\end{minipage}}\end{center}Suppose that the inequality$0 < \sum_{n=1}^{\infty} \left(b_n - \frac{1}{a_n a_{n+1}}\right) < 30$holds. Then$a_2 \times \sum_{n=1}^{\infty} b_{2n} = \dfrac{q}{p},$where $a_1 eq 0$ and $p$ and $q$ are coprime natural numbers. Compute $p + q$. [4 points]\