$S = \dfrac{1}{2} + \dfrac{1}{2.3} + \dfrac{1}{3.4} + .... + \dfrac{1}{99.100}$
$⇔S = \dfrac{1}{1.2} + \dfrac{1}{2.3} + \dfrac{1}{3.4} + .... + \dfrac{1}{99.100}$
$⇔S = 1 - \dfrac{1}{2} + \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + ..... + \dfrac{1}{99} - \dfrac{1}{100}$
$⇔S = 1 - \dfrac{1}{100}$
$⇔S = \dfrac{99}{100}$