$\dfrac{1}{2.3} + \dfrac{1}{3.4} + \dfrac{1}{4.5} + ..... + \dfrac{1}{98.99} + \dfrac{1}{99.100}$
$= \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + \dfrac{1}{4} - \dfrac{1}{5} + ..... + \dfrac{1}{98} - \dfrac{1}{99} + \dfrac{1}{99} - \dfrac{1}{100}$
$= \dfrac{1}{2} - \dfrac{1}{100}$
$= \dfrac{49}{100}$