TITLE: Show that $\displaystyle\sum\limits_{n = 2}^\infty \frac{1}{f_{n-1}\,f_{n+1}} = 1$ by the telescoping technique
QUESTION [0 upvotes]: I'm not sure how to approach or set this up. Thanks

REPLY [3 votes]: $$
f_{n+1} = f_{n}+f_{n-1} \rightarrow f_{n} = f_{n+1}-f_{n-1}\\
\frac{1}{f_{n-1}f_{n+1}} = \left(\frac{1}{f_{n-1}} - \frac{1}{f_{n+1}}\right)\cdot \frac{1}{f_n}
$$
$$
\sum_{n=2}^{\infty}\frac{1}{f_{n-1}f_{n+1}} = \sum_{n=2}^{\infty}\left(\frac{1}{f_{n}f_{n-1}} - \frac{1}{f_{n+1}f_{n}}\right) = \frac{1}{1\cdot 1} = 1
$$
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