Question

$\frac{1}{3\times 7}+\frac{1}{7\times 11}+\frac{1}{11\times 15}+......\frac{1}{47\times 51}$

• A. $\frac{16}{51}$
• B. $\frac{8}{51}$
• C. $\frac{4}{51}$
• D. $\frac{2}{51}$

Answer 1

The correct option is

C

$\frac{4}{51}$

Solution :-

$\frac{1}{3\times 7}+\frac{1}{7\times 11}+\frac{1}{11\times 15}+...$

We have $\frac{1}{3\times 7}=\frac{7-3}{7\times 3}=\frac{1}{4}\left[\frac{4}{7\times 3}\right]=\frac{1}{4\times 3}-\frac{1}{4\times 7}$

Similarly $\frac{1}{7\times 11}=\frac{1}{4}\left[\frac{11-7}{7\times 11}\right]=\frac{1}{7\times 4}-\frac{1}{4\times 11}$

$\therefore$ summing all terms we have

$\frac{1}{3\times 7}+\frac{1}{7\times 11}+\frac{1}{11\times 15}+...+\frac{1}{47\times 51}$

$=\frac{1}{4\times 3}-\frac{1}{4\times 7}+\frac{1}{7\times 4}-\frac{1}{4\times 11}+...+\frac{1}{4\times 47}-\frac{1}{4\times 51}$

$=\frac{1}{4\times 3}-\frac{1}{4\times 51}$

$=\frac{4}{51}$