A) \[\frac{5n-4}{5n+6}\]
B) \[\frac{n}{(5n+6)}\]
C) \[\frac{2n-1}{5n+6}\]
D) \[\frac{1}{(5n+6)}\]
Correct Answer: B
Solution :
Given that, \[{{S}_{n}}=\frac{1}{6.11}+\frac{1}{11.16}+\frac{1}{16.21}+....+n\]terms \[=\frac{1}{5}\left( \frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+....\frac{1}{5n+1}-\frac{1}{5n+6} \right)\] \[=\frac{1}{5}\left( \frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+....+\frac{1}{5n+1}-\frac{1}{5n+6} \right)\] \[=\frac{1}{5}\left( \frac{1}{6}-\frac{1}{5n+6} \right)\] \[=\frac{1}{5}\left( \frac{5n+6-6}{6(5n+6)} \right)\] \[=\frac{n}{6(5n+6)}\] \[\Rightarrow \] \[6{{S}_{n}}=\frac{n}{5n+6}\]
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