16 The sum of $\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{13 \times 14 \times 15}+\frac{1}{14 \times 15 \times 16}$ is $\frac{m}{n}$ in its lowest terms. Find the value of $m+n$.
16 Answer: (173)
Note that $\frac{1}{n \times(n+1) \times(n+2)}=\frac{1}{2}\left(\frac{1}{n \times(n+1)}-\frac{1}{(n+1) \times(n+2)}\right)$.
Hence $\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{13 \times 14 \times 15}+\frac{1}{14 \times 15 \times 16}$ $=\frac{1}{2}\left(\frac{1}{2 \times 3}-\frac{1}{15 \times 16}\right)=\frac{13}{160}$.