Finding the n-th partial sum of a series

Question

I am trying to find the partial sum of the series defined by:

$$\frac{9}{1.2.3}+\frac{9}{2.3.4}+\frac{9}{3.4.5}+....+\frac{9}{n(n+1)(n+2)}+...$$

I know the answer is $$\frac{9n(n+3)}{4(n+1)(n+2)}$$t

I tried the telescopic series but didn't work. Any suggestions?

Answer 1

Use partial fractions to get $$a_n=\frac92 \left (\frac1{r}-\frac1{r+1} \right )-\frac92 \left (\frac1{r+1}-\frac1{r+2} \right )$$ Now use telescoping.