Question 156
The sequences of real numbers $\{a_{i}\}_{i=1}^{\infty}$ and $\{b_{i}\}_{i=1}^{\infty}$ satisfy $a_{n+1} = (a_{n-1} - 1)(b_{n} + 1)$ and $b_{n+1} = a_{n} b_{n-1} - 1$ for $n \geq 2$, with $a_{1} = a_{2} = 2015$ and $b_{1} = b_{2} = 2013$. Evaluate, with proof, the infinite sum
$$
\sum_{n=1}^{\infty} b_{n}\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n+3}}\right)
$$
$$
\sum_{n=1}^{\infty} b_{n}\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n+3}}\right)
$$