(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,

\[{{S}_{n}}~=\text{ }a(1\text{ }-\text{ }{{r}^{n~}})/\text{ }1\text{ }-\text{ }r\]