Q. If $S_{n} = \frac{1}{6.11} + \frac{1}{11.16} + \frac{1}{16.21} + ...to\ n$ terms, then $6S_{n}$ equals
A) $\frac{5n - 4}{5n + 6}$
B) $\frac{n}{(5n + 6)}$
C) $\frac{2n - 1}{5n + 6}$
D) $\frac{1}{(5n + 6)}$
Explanation
(b)
Let $S_{n} = \frac{1}{5}\left( \frac{1}{6} - \frac{1}{11} + \frac{1}{11} - \frac{1}{16} + ... + \frac{1}{5n + 1} - \frac{1}{5n + 6} \right)$
$= \frac{1}{5}\left( \frac{1}{6} - \frac{1}{11} + \frac{1}{11} - \frac{1}{16} + ... + \frac{1}{5n + 1} - \frac{1}{5n + 6} \right)$
$= \frac{1}{5}\left( \frac{1}{6} - \frac{1}{5n + 6} \right) = \frac{n}{6(5n + 6)}$
$\Rightarrow 6S_{n} = \frac{n}{5n + 6}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.