$$\frac{1}{{20}}$$ $$ + $$ $$\frac{1}{{30}}$$ $$ + $$ $$\frac{1}{{42}}$$ $$ + $$ $$\frac{1}{{56}}$$ $$ + $$ $$\frac{1}{{72}}$$ $$ + $$ $$\frac{1}{{90}}$$ $$ + $$ $$\frac{1}{{110}}$$  $$ + $$ $$\frac{1}{{132}}$$  is equal to :

Solution(By Examveda Team)

$$\frac{1}{{20}} + \frac{1}{{30}} + \frac{1}{{42}} + \frac{1}{{56}} + \frac{1}{{72}} + \frac{1}{{90}}$$      $$ + \frac{1}{{110}}$$   $$ + \frac{1}{{132}}$$
$$ = \left( {\frac{1}{{4 \times 5}}} \right) + \left( {\frac{1}{{5 \times 6}}} \right) + \left( {\frac{1}{{6 \times 7}}} \right)$$       $$ + ..... + $$   $$\left( {\frac{1}{{11 \times 12}}} \right)$$
$$ = \left( {\frac{1}{4} - \frac{1}{5}} \right) + \left( {\frac{1}{5} - \frac{1}{6}} \right) + \left( {\frac{1}{6} - \frac{1}{7}} \right)$$       $$ + ..... + $$   $$\left( {\frac{1}{{11}} - \frac{1}{{12}}} \right)$$
$$ = \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6} + \frac{1}{6} - \frac{1}{7} \, + .....$$      $$ + \frac{1}{{11}}$$  $$ - \frac{1}{{12}}$$
$$\eqalign{ & = \frac{1}{4} - \frac{1}{{12}} \cr & = \frac{{3 - 1}}{{12}} \cr & = \frac{2}{{12}} \cr & = \frac{1}{6} \cr} $$