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

${\mathrm{T}}_{10}=\frac{1}{20}=\mathrm{a}+9\mathrm{d}---\left(1\right)$

$\begin{array}{l}{\mathrm{T}}_{20}=\frac{1}{100}=\mathrm{a}+19\mathrm{d}---\left(2\right)\\ \left(1\right)\text{ and }\left(2\right)\Rightarrow \text{ we get }\mathrm{a}=\frac{1}{200},\mathrm{d}=\frac{1}{200}\\ {\mathrm{S}}_{200}=\frac{200}{2}\left[\frac{2}{200}+\frac{199}{200}\right] \left(sum of n terms in A.P=\frac{n}{2}\left(2a+\left(n-1\right)d\right)\right)\\ =\frac{201}{2}=100.5\end{array}$