(i)
(ii)
Solution:
(i) According to the given question
Here,
\[a\text{ }=\text{ }\left( x\text{ }+\text{ }y \right)/\text{ }\left( x\text{ }-\text{ }y \right)\]
And
\[~r\text{ }=\text{ }1/\left[ \left( x\text{ }+\text{ }y \right)/\text{ }\left( x\text{ }-\text{ }y \right) \right]~\]
\[=\text{ }\left( x\text{ }-\text{ }y \right)/\text{ }\left( x\text{ }+\text{ }y \right)\text{ }\left( \left| \text{ }r\text{ } \right|\text{ }<\text{ }1 \right)\]
Number of terms \[=\text{ }n\]
Thus,
\[{{S}_{n}}~=\text{ }a(1\text{ }-\text{ }{{r}^{n~}})/\text{ }1\text{ }-\text{ }r\]
(ii) According to the given question
Here,
\[a\text{ }=\text{ }\surd 3\]
And
\[r\text{ }=\text{ }1/\surd 3/\text{ }\surd 3\text{ }=\text{ }1/3\text{ }\left( \left| \text{ }r\text{ } \right|\text{ }<\text{ }1 \right)\]
Number of terms \[=\text{ }n\]
Thus,
\[{{S}_{n}}~=\text{ }a(1\text{ }-\text{ }{{r}^{n~}})/\text{ }1\text{ }-\text{ }r\]