Question

Evaluate:
$\frac{1}{3\times 7\times 11}+\frac{1}{7\times 11\times 15}+\frac{1}{11\times 15\times 19}+...$ upto $20$ terms.

Answer 1

$S=\frac{1}{8}[\frac{11-3}{\mathrm{3.7.11}}+\frac{15-7}{\mathrm{7.11.15}}+\frac{19-11}{\mathrm{11.15.19}}]$

$=\frac{1}{8}[\frac{1}{3.7}-\frac{1}{7.11}+\frac{1}{7.11}-\frac{1}{11.15}+\frac{1}{11.15}-\frac{1}{15.19}+.....]-\frac{1}{83.87}$

For ${20}^{th}$ term, $3,7,11,.....\left(20\right)$

$a=3,d=4$

$a_{20}=3+19\left(9\right)=79$

$\Rightarrow {20}^{th}$ term $=\frac{1}{79\times 83\times 87}$

$\Rightarrow S=\frac{1}{8}(\frac{1}{21}-\frac{1}{7221})=\frac{1}{8}\times \frac{7200}{21\times 7221}=\frac{300}{7\times 7221}=\frac{300}{5057}$

$\Rightarrow S=\frac{100}{16849}$.