<s>[INST] Calculate $ S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}. $ Calculate $S =\displaystyle\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. This sequence is neither arithmetic nor geometric. How can you solve this. Thanks!
 [/INST] Since $$\frac{1}{k(k+2)} = \frac{1}{2} \frac{2}{k(k+2)} = \frac{1}{2}\left(\frac{1}{k} - \frac{1}{k+2}\right)$$
for all $k$, we have
$$\sum_{k = 1}^n \frac{1}{k(k+1)(k+2)} = \frac{1}{2}\sum_{k = 1}^n \left(\frac{1}{k(k+1)} - \frac{1}{(k+1)(k+2)}\right),$$
which telescopes to $$\frac{1}{2}\left(\frac{1}{2} - \frac{1}{(n+1)(n+2)}\right) = \frac{1}{4} - \frac{1}{2(n+1)(n+2)}.$$
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