Show that $\displaystyle\sum\limits_{n = 2}^\infty \frac{1}{f_{n-1}\,f_{n+1}} = 1$ by the telescoping technique

Question

I'm not sure how to approach or set this up. Thanks

Answer 1

$$ f_{n+1} = f_{n}+f_{n-1} \rightarrow f_{n} = f_{n+1}-f_{n-1}\\ \frac{1}{f_{n-1}f_{n+1}} = \left(\frac{1}{f_{n-1}} - \frac{1}{f_{n+1}}\right)\cdot \frac{1}{f_n} $$ $$ \sum_{n=2}^{\infty}\frac{1}{f_{n-1}f_{n+1}} = \sum_{n=2}^{\infty}\left(\frac{1}{f_{n}f_{n-1}} - \frac{1}{f_{n+1}f_{n}}\right) = \frac{1}{1\cdot 1} = 1 $$