Question

If the ${10}^{th}$ term of an A.P. is $\frac{1}{20}$ and its ${20}^{th}$ term is $\frac{1}{10}$, then the sum of its first $200$ terms is

• A. $99$
• B. $100$
• C. $101$
• D. $100\frac{1}{2}$

Answer 1

The correct option is D $100\frac{1}{2}$

Given: $T_{10}=\frac{1}{20},T_{20}=\frac{1}{10}$

$\Rightarrow T_{20}-T_{10}=10d$
$\Rightarrow \frac{1}{20}=10d$
$\Rightarrow d=\frac{1}{200}$

Now, $T_{10}=\frac{1}{20}$
$\Rightarrow a+9d=\frac{1}{20}$
$\Rightarrow a=\frac{1}{200}$

$S_{200}=\frac{200}{2}[2\left(\frac{1}{200}\right)+199\left(\frac{1}{200}\right)]$
$\therefore S_{200}=100\frac{1}{2}$