Question

Suppose $f$ is a function satisfying $f(x+y)=f(x)+f(y)$ for all $x, y \in N$ and $f(1)=\frac{1}{5}$. If $\displaystyle\sum_{n=1}^{\text {m }} \frac{f(n)}{n(n+1)(n+2)}=\frac{1}{12}$, then $m$ is equal to_____

Answer 1

$ \because f(1)=\frac{1}{5} \therefore f(2)=f(1)+f(1)=\frac{2}{5} $
$ f(2)=\frac{2}{5} f(3)=f(2)+f(1)=\frac{3}{5}$
$f(3)=\frac{3}{5} $
$ \therefore \displaystyle\sum_{n=1}^m \frac{f(n)}{n(n+1)(n+2)} $
$=\frac{1}{5} \displaystyle\sum_{n=1}^m\left(\frac{1}{n+1}-\frac{1}{n+2}\right) $
$=\frac{1}{5}\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\ldots .+\frac{1}{m+1}-\frac{1}{m+2}\right)$
$ =\frac{1}{5}\left(\frac{1}{2}-\frac{1}{m+2}\right)=\frac{m}{10(m+2)}=\frac{1}{12} $
$ \therefore m=10$