$B = \dfrac{25}{1.2.3} + \dfrac{25}{2.3.4} + \dfrac{25}{3.4.5} + ... + \dfrac{25}{20.21.22}$
$\to B = \dfrac{25}{2}.\left ( \dfrac{2}{1.2.3} + \dfrac{2}{2.3.4} + \dfrac{2}{3.4.5} + ... + \dfrac{2}{20.21.22} \right )$
$\to B = \dfrac{25}{2}.\left [ \dfrac{1}{2}.\left ( \dfrac{2}{1.3} \right ) + \dfrac{1}{3}.\left ( \dfrac{2}{2.4} \right ) + \dfrac{1}{4}.\left ( \dfrac{2}{3.5} \right ) + ... + \dfrac{1}{21}.\left ( \dfrac{2}{20.22} \right ) \right ]$
$\to B = \dfrac{25}{2}.\left [ \dfrac{1}{2}.\left ( 1 - \dfrac{1}{3} \right ) + \dfrac{1}{3}.\left ( \dfrac{1}{2} - \dfrac{1}{4} \right ) + \dfrac{1}{4}.\left ( \dfrac{1}{3} - \dfrac{1}{5} \right ) + ... + \dfrac{1}{21}.\left ( \dfrac{1}{20} - \dfrac{1}{222} \right ) \right ]$
$\to B = \dfrac{25}{2}.\left ( \dfrac{1}{1.2} - \dfrac{1}{2.3} + \dfrac{1}{2.3} - \dfrac{1}{3.4} + \dfrac{1}{3.4} - \dfrac{1}{4.5} + ... + \dfrac{1}{20.21} - \dfrac{1}{21.22} \right )$
$\to B = \dfrac{25}{2}.\left ( \dfrac{1}{1.2} - \dfrac{1}{21.22} \right )$
$\to B = \dfrac{2875}{462}$
