# Calculate sum in a short way

The question is like this: Let the sum $$\sum_{n=1}^9 \frac{1}{n(n+1)(n+2)}$$ written in its lowest terms be $\frac{p}{q}$. Find $p-q$ I tried to calculate it by putting in values 1 to 9 and actually calculating the value of the sum, but it was too long and I don't want to use a calculator. I messed with the expression for quite a while only to realize it was in vain. Please help. thanks in advance.

• $\phantom{}$TLSCPC – Jack D'Aurizio Dec 28 '17 at 17:43
• @ Jack D'Aurizio: aramaic-english? – cgiovanardi Dec 29 '17 at 13:48

Hint: $$\frac1{n(n+1)(n+2)}=\frac12\left[\frac1{n(n+1)}-\frac1{(n+1)(n+2)}\right]$$
• Yes, that is correct up to a factor of $-1$. – robjohn Dec 28 '17 at 17:08