What is the value of $$\frac{7}{2} + \frac{{11}}{3} + \frac{7}{6} + \frac{{11}}{{15}} + \frac{7}{{12}} + \frac{{11}}{{35}} + ........ + \frac{7}{{156}} + \frac{{11}}{{575}}$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{7}{2} + \frac{{11}}{3} + \frac{7}{6} + \frac{{11}}{{15}} + \frac{7}{{12}} + \frac{{11}}{{35}} + \,........\, + \frac{7}{{156}} + \frac{{11}}{{575}} \cr & = \left( {\frac{7}{2} + \frac{7}{6} + \frac{7}{{12}} + \,........\, + \frac{7}{{156}}} \right) + \left( {\frac{{11}}{3} + \frac{{11}}{{15}} + \frac{{11}}{{35}} + \,........\, + \frac{{11}}{{575}}} \right) \cr & = 7\left( {\frac{1}{{1 \times 2}} + \frac{1}{{2 \times 3}} + \frac{1}{{3 \times 4}} + \,........\, + \frac{1}{{12 \times 13}}} \right) + \frac{{11}}{2}\left( {\frac{2}{{1 \times 3}} + \frac{2}{{3 \times 5}} + \frac{2}{{5 \times 7}} + \,........\, + \frac{1}{{23 \times 25}}} \right) \cr & = 7\left( {1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \,........\, + \frac{1}{{12}} - \frac{1}{{13}}} \right) + \frac{{11}}{2}\left( {1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \,........\, + \frac{1}{{23}} - \frac{1}{{25}}} \right) \cr & = 7\left( {1 - \frac{1}{{13}}} \right) + \frac{{11}}{2}\left( {1 - \frac{1}{{25}}} \right) \cr & = \frac{{7 \times 12}}{{13}} + \frac{{11 \times 24}}{{2 \times 25}} \cr & = \frac{{84}}{{13}} + \frac{{132}}{{25}} \cr & = \frac{{2100 + 1716}}{{325}} \cr & = \frac{{3816}}{{325}} \cr} $$