Calculate sum in a short way

Question

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.

@ Jack D'Aurizio: aramaic-english? – cgiovanardi Dec 29 '17 at 13:48 — cgiovanardi, Dec 29 '17 at 13:48

score 4 · Accepted Answer · answered Dec 28 '17 at 16:56

4

Hint: $$ \frac1{n(n+1)(n+2)}=\frac12\left[\frac1{n(n+1)}-\frac1{(n+1)(n+2)}\right] $$

answered Dec 28 '17 at 16:56

robjohn

345,667

Is the answer 83? – user167920 Dec 28 '17 at 17:06
Yes, that is correct up to a factor of $-1$. – robjohn Dec 28 '17 at 17:08
Oh yeah right it was numerator minus denominator. -83. My bad. – user167920 Dec 28 '17 at 17:17
I like this. I usually decompose it into 3 fractions. Never thought of doing this. – Higurashi Dec 28 '17 at 17:36

Calculate sum in a short way

1 Answers1