# Looking for $\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$

I am looking for the sum of the series:

$$\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$$

where $F_{n}$ is the $n$-th Fibonacci number. I was thinking about splitting the fraction into 2 like in the case of $\dfrac{1}{n(n+1)}$, but do not see how it should work in this particular case. Thank you.

\begin{align}\sum_{n=1}^{N}\frac{1}{f(n)f(n+2)} &=\sum_{n=1}^{N}\frac{\color{red} {f(n+1)}}{f(n)\color{red} {f(n+1)}f(n+2)}\\ &=\sum_{n=1}^{N}\frac{f(n+2)-f(n)}{f(n)f(n+1)f(n+2)}\\ &=\sum_{n=1}^{N}\frac{1}{f(n)f(n+1)}-\frac{1}{f(n+1)f(n+2)}\\ &=\frac{1}{f(1)f(2)}-\frac{1}{f(N+1)f(N+2)}\end{align}.