Correct option is (3) 1/5
\(S_n = \frac7{1^2 .6^2}+ \frac{17}{6^2.11^2} + \frac{27}{11^2.16^2}+....+\frac{10n -3}{(5n - 4)^2 (5n + 1)^2} + ....\)
\(= \frac{1+6}{1^2.6^2}+\frac{6+11}{6^2.11^2} + \frac{11 + 16}{11^2.16^2} + ....+\frac{(5n - 4) + (5n + 1)}{(5n - x)^2 (5n+1)^2}+...\)
\(=\frac15\left( \frac{6^2-1^2}{6^2.1^2}+\frac{11^2-6^2}{11^2.6^2} + \frac{16^2 - 11^2}{16^2 . 11^2} + ....+\frac{(5n +1)^2 - (5n + 4)^2}{(5n - 4)^2 (5x+1)^2}+...\right)\)
\(= \frac15 \left( \left(\frac1{1^2}- \frac{1}{6^2}\right) +\left(\frac1{6^2}-\frac1{11^2}\right)+ \left(
\frac1{11^2}-\frac1{16^2}\right)+...+ \left(\frac1{(5n - 1)^2} - \frac1{(5n +1)^2}\right)+... \right)\)
\(= \frac15\left(\frac1{12}- \frac1\infty\right)\)
\(= \frac15 (1 - 0) = \frac15\)