I am having some trouble proving following identity without use of induction, with which it is trivial.

$$\sum\_{n=1}^{m}\frac{1}{n(n+1)(n+2)}=\frac{1}{4}-\frac{1}{2(m+1)(m+2)}$$

I did expand the expression:
$$\sum\_{n=1}^{m}\left( \frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2(n+2)} \right)$$

I have no idea how to proceed further.