Official solution
T10 = a + 9d = \[{1 \over {20}}\] ....(1)
T20 = a + 19d = \[{1 \over {10}}\] .....(2)
Equation (2) -- (1)
10d = \[{1 \over {10}}\] - \[{1 \over {20}}\]
\[ \Rightarrow \] d = \[{1 \over {200}}\]
a + \[{9 \over {200}}\] = \[{1 \over {20}}\]
\[ \Rightarrow \] a = \[{1 \over {200}}\]
S200 = \[{{200} \over 2}\left[ {{2 \over {200}} + \left( {200 - 1} \right) \times {1 \over {200}}} \right]\]
= \[100\left[ {{2 \over {200}} + {{199} \over {200}}} \right]\]
= \[{{201} \over 2}\] = \[100{1 \over 2}\]