1)

The sum of the first 20 terms of the series

$\frac{1}{5 \times 6} + \frac{1}{6 \times 7}+\frac{1}{7 \times 8}+....$ is

A) 0.16

B) 1.6

C) 16

D) None of these

Option A

### Explanation:

The series is $\sum_1^\infty\frac{1}{(n+4)(n+5)}$ ;n = 1,2,3,4...20
we can write the above as:

$\sum_1^\infty\frac{1}{(n+4)}-\frac{1}{n+5}$

20th term is
$\frac{1}{(20+4)}-\frac{1}{(20+5)}$

=$\frac{1}{24}-\frac{1}{25}$
Then the sum of 20 terms of series is
$\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{8}\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\frac{1}{23}-\frac{1}{24}+\frac{1}{24}-\frac{1}{25}$
cancel the plus minus we get as the below:
$\frac{1}{5}-\frac{1}{25}$
$\frac{4}{25}$
= 0.16