A 2012 Problem

$S = \dfrac{1}{2\times 5} + \dfrac{1}{5\times 8} + \dfrac{1}{8\times 11} + \ldots + \dfrac{1}{2009\times 2012}$

$S = \displaystyle \sum_{n=0}^{669} \dfrac{1}{(3n+2)\times (3n+5)} = \dfrac{1}{3} \sum_{n=0}^{669} \dfrac{1}{3n+2} - \dfrac{1}{3n+5}$

$S = \dfrac{1}{3}\left(\dfrac{1}{2} - \dfrac{1}{2012}\right)$

$S = \boxed{\dfrac{335}{2012}}$