Problem 88E

88. The Fibonacci sequence was defined in Section $11.1$ by the equations

$$f_{1}=1, \quad f_{2}=1, \quad f_{n}=f_{n-1}+f_{n-2} \quad n \geqslant 3$$

Show that each of the following statements is true.

(a) $\frac{1}{f_{n-1} f_{n+1}}=\frac{1}{f_{n-1} f_{n}}-\frac{1}{f_{n} f_{n+1}}$

(b) $\sum_{n=2}^{\infty} \frac{1}{f_{n-1} f_{n+1}}=1$

(c)  $\sum_{n=2}^{\infty} \frac{f_{n}}{f_{n-1} f_{n+1}}=2$

Step-by-Step Solution