Prove the following


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

  1. (1) 50

  2. (2) $50 \frac{1}{4}$

  3. (3) 100

  4. (4) $100 \frac{1}{2}$

Correct Option: , 4


$T_{10}=\frac{1}{20}=a+9 d$...(i)

$T_{20}=\frac{1}{10}=a+19 d$...(ii)

Solving equations (i) and (ii), we get

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

$\Rightarrow \quad S_{200}=\frac{200}{2}\left[\frac{2}{200}+\frac{199}{200}\right]=\frac{201}{2}=100 \frac{1}{2}$

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