I need to prove several proofs related to the Fibonacci sequence and I don't have the faintest clue how to do so. Please help!
Given that the Fibonacci sequence is defined as $f_n = f_{n-1} + f_{n-2}$ for $n \ge 3 $
I need to prove that:
$$\frac 1{f_{n-1}f_{n+1}} = \frac 1{f_{n-1}f_n} - \frac 1{f_nf_{n+1}}$$
and that
$$\sum_{n=2}^{\infty} \frac 1 {f_{n-1}f_{n+1}} = 1$$
and that
$$\sum_{n=2}^{\infty} \frac {f_n} {f_{n-1}f_{n+1}} = 2$$
Any help whatsoever you could give me would be appreciated. Even just hints as how to approach these problems. Anything you can give me.
Hint: rewrite $f_{n+1} = f_{n} + f_{n-1}$ for all 3.
First equation: $$\frac{1}{f_{n-1}(f_n f_{n-1})} = \frac{1}{f_{n-1}f_n } - \frac{1}{f_n(f_n + f_{n-1})}$$
Third equation, use:
$$\frac{f_n}{f_{n-1}f_{n+1}} = \frac{1}{f_{n-1}}-\frac{1}{f_{n+1}}$$