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!

$$u_k = \dfrac{1}{k(k+1)(k+2)} = \dfrac{1}{2k(k+1)} - \dfrac{1}{2(k+1)(k+2)} = 
f_k - f_{k+1}$$
so $$u_1 + u_2 + \cdots u_n = (f_1 - f_2) + (f_2 - f_3) + \cdots + (f_n - f_{n+1}) = f_1 - f_{n+1} = 
\dfrac{1}{4} - \dfrac{1}{2(n+1)(n+2)} 
$$
