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Question: The sum of the series \(\frac{1}{3 \times 7} + \frac{1}{7 \times 11} + \frac{1}{11 \times 15} + .......

The sum of the series 13×7+17×11+111×15+....\frac{1}{3 \times 7} + \frac{1}{7 \times 11} + \frac{1}{11 \times 15} + .... is

A

13\frac{1}{3}

B

16\frac{1}{6}

C

19\frac{1}{9}

D

112\frac{1}{12}

Answer

112\frac{1}{12}

Explanation

Solution

S=(13×7+17×11+111×15+....)=14[(13−17)+(17−111)+(111−115)+.......1∞]S = \left( \frac{1}{3 \times 7} + \frac{1}{7 \times 11} + \frac{1}{11 \times 15} + .... \right) = \frac{1}{4}\left\lbrack \left( \frac{1}{3} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{11} \right) + \left( \frac{1}{11} - \frac{1}{15} \right) + .......\frac{1}{\infty} \right\rbrack =14[13−1∞]=14[13−0]=112= \frac{1}{4}\left\lbrack \frac{1}{3} - \frac{1}{\infty} \right\rbrack = \frac{1}{4}\left\lbrack \frac{1}{3} - 0 \right\rbrack = \frac{1}{12}