$A = \dfrac{1}{1.2.3} + \dfrac{1}{2.3.4} + \dfrac{1}{3.4.5} + ... + \dfrac{1}{18.19.20}$
$\to A = \dfrac{1}{2}.\left ( \dfrac{2}{1.2.3} + \dfrac{2}{2.3.4} + \dfrac{2}{3.4.5} + ... + \dfrac{2}{18.19.20} \right )$
$\to A = \dfrac{1}{2}.\left [ \dfrac{1}{2}.\left ( \dfrac{2}{1.3} \right ) + \dfrac{1}{3}.\left ( \dfrac{2}{2.4} \right ) + \dfrac{1}{4}.\left ( \dfrac{2}{3.5} \right ) + ... + \dfrac{1}{19}.\left ( \dfrac{2}{18.20} \right ) \right ]$
$\to A = \dfrac{1}{2}.\left [ \dfrac{1}{2}.\left ( 1 - \dfrac{1}{3} \right ) + \dfrac{1}{3}.\left ( \dfrac{1}{2} - \dfrac{1}{4} \right ) + \dfrac{1}{4}.\left ( \dfrac{1}{3} - \dfrac{1}{5} \right ) + ... + \dfrac{1}{19}.\left ( \dfrac{1}{18} - \dfrac{1}{20} \right ) \right ]$
$\to A = \dfrac{1}{2}.\left ( \dfrac{1}{1.2} - \dfrac{1}{2.3} + \dfrac{1}{2.3} - \dfrac{1}{3.4} + \dfrac{1}{3.4} - \dfrac{1}{4.5} + ... + \dfrac{1}{18.19} - \dfrac{1}{19.20} \right )$
$\to A = \dfrac{1}{2}.\left ( \dfrac{1}{1.2} - \dfrac{1}{19.20} \right )$
$\to A = \dfrac{1}{4} - \dfrac{1}{760} < \dfrac{1}{4}$
