For each positive integer $n$, let $$S_n= \frac1{n(n+1)} + \frac1{(n+1)(n+2)} +\dots + \frac1{(2n−1)2n}.$$
(a) Calculate
$S_1,S_2,S_3$. Then use this data to guess a simple formula for
$S_n$.
(b) Prove your guess in part (a) by mathematical induction.
(c) Use Result 6.6 on page 136 to give another proof of your guess.
(d) Prove that
$$\frac1{k(k+1)}=\frac1k−\frac1{k+1}$$
for all positive real numbers $k$. Use this to give yet another proof of your guess in part (a). This method of proof is called telescoping.
Result 6.6: For every positive integer $n$: $$\frac1{(2)(3)} + \frac1{(3)(4)} +\dots + \frac1{(n+1)(n+2)} = \frac{n}{2n+4}.$$
-
I've already given a proof of 2(b) for that $S_{n} = \frac1{2n}$, but I am stuck on (c) and (d).