Q64 / 180 · COMEDK 2016 · Mathematics

If $S_{n}=\frac{1}{6 \cdot 11}+\frac{1}{11 \cdot 16}+\frac{1}{16 \cdot 21}+\ldots$ to $n$ terms, then $6 S_{n}=$

  • A
    $\frac{1}{(5 n+6)}$
  • B
    $\frac{(2 n-1)}{5 n+6}$
  • Correct answer
    C
    $\frac{n}{(5 n+6)}$
  • D
    $\frac{5 n-4}{5 n+6}$

Solution

We have, $S_{n}=\frac{1}{6 \cdot 11}+\frac{1}{11 \cdot 16}+\frac{1}{16 \cdot 21}+\ldots+n \text { terms }$ $\quad=\frac{1}{5}\left(\frac{1}{6}-\frac{1}{11}+\frac{1}{11}-\frac{1}{16}+\ldots \frac{1}{5 n-1}-\frac{1}{5 n+6}\right)$ $=\frac{1}{5}\left(\frac{1}{6}-\frac{1}{5 n+6}\right)$ $=\frac{1}{5}\left[\frac{5 n+6-6}{6(5 n+6)}\right]$ $=\frac{1}{5}\left[\frac{5 n}{6(5 n+6)}\right]$ So, $6 \mathrm{~S}_{n}=\frac{n}{(5 n+6)}$

Hence, the correct answer is option: 'C'.

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