Question

The sum of first $20$ terms of
$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}........$ is

• A. $\frac{21}{20}$
• B. $\frac{20}{21}$
• C. $\frac{19}{20}$
• D. $\frac{20}{19}$

Answer 1

Answer: (B)

$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}...$
$T_n=\frac{1}{n\left(n+1\right)}=\frac{1}{n}-\frac{1}{n+1}$
$\therefore T_1=1-\frac{1}{2}$
$T_2=\frac{1}{2}-\frac{1}{3}$
$T_3=\frac{1}{3}-\frac{1}{4}$
$T_{20}=\frac{1}{20}-\frac{1}{21}$
On adding, we get
$S_{20}=1-\frac{1}{21}=\frac{20}{21}$