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. $50\frac{1}{4}$
• B. $100$
• C. $100\frac{1}{2}$
• D. $50$

Answer 1

The correct option is C

$100\frac{1}{2}$

Explanation for Correct option:

Step 1: Apply the A.P. formula

​Let first term =a

common difference =d

${\mathrm{T}}_{10}=\frac{1}{20}$
$\Rightarrow a+9d=\frac{1}{20}$ ……..(i)

${\mathrm{T}}_{20}=\frac{1}{10}$
$\Rightarrow 9+19d=\frac{1}{10}$​ ……..(ii)
Step 2:On solving equations (i) and (ii)

$a=\frac{1}{200},d=\frac{1}{200}$

$\therefore S_{200}=\frac{200}{2}\left[\frac{2}{200}+(200-1)\frac{1}{200}\right]$
$\therefore S_{200}=\frac{201}{2}$
$\therefore S_{200}=100\frac{1}{2}$

Hence Option (4) is Correct