Series of Fractional Sums

Question

What is the sum of $$\frac{1}{2\times 5} + \frac{1}{5\times 8}+ \frac{1}{8\times 11}+...+ \frac{1}{2009\times 2012}?$$

What is the easiest way to solve this kind of problem?

Answer 1

HINT:

$$\frac1{r(r+3)}=\frac13\left(\frac1r-\frac1{r+3}\right)$$

It leaves behind a nice telescopic sum