a) $\ \dfrac{1}{2.3.4} +\dfrac{1}{3.4.5} +...+ \dfrac{22.23.24}$
$ = \dfrac{1}{2}(\dfrac{2}{2.3.4} \dfrac{2}{3.4.5} + ...+\dfrac{22.23.24} $
$\ = \dfrac{1}{2}(\dfrac{1}{2.3} - \dfrac{1}{3.4} + \dfrac{1}{3.4} - \dfrac{4.5} + ...+\dfrac{22.23} - \dfrac{1}{23.24}$
$\ = \dfrac{1}{2}(\dfrac{1}{6} - \dfrac{1}{552}$
$\ = \dfrac{91}{552}$