$A=\dfrac{1}{2}.\dfrac{1}{3} + \dfrac{1}{3}.\dfrac{1}{4} + \dfrac{1}{4}.\dfrac{1}{5} +......+ \dfrac{1}{8}.\dfrac{1}{9}$
$⇔ A= \dfrac{1}{2.3}+\dfrac{1}{3.4} + \dfrac{1}{4.5}+ .......+ \dfrac{1}{8.9}$
$⇔ A = \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + \dfrac{1}{4} - \dfrac{1}{5} + ..... + \dfrac{1}{8} - \dfrac{1}{9}$
$⇔ A = \dfrac{1}{2} - \dfrac{1}{9}$
$⇔ A = \dfrac{7}{18}$
