A) \[n-\frac{1}{2}({{3}^{n}}-1)\]
B) \[n+\frac{1}{2}({{3}^{n}}-1)\]
C) \[n+\frac{1}{2}(1-{{3}^{-n}})\]
D) \[n+\frac{1}{2}({{3}^{-n}}-1)\]
Correct Answer: D
Solution :
\[{{T}_{n}}=\frac{{{3}^{n}}-1}{{{3}^{n}}}=1-{{\left( \frac{1}{3} \right)}^{n}}\] \[{{S}_{n}}=n-\sum\limits_{n=1}^{n}{{{\left( \frac{1}{3} \right)}^{n}}}\]\[=n-\frac{\frac{1}{3}\left[ 1-{{\left( \frac{1}{3} \right)}^{n}} \right]}{\left( 1-\frac{1}{3} \right)}\] \[=n-\frac{1}{2}(1-{{3}^{-n}})=n+\frac{1}{2}({{3}^{-n}}-1)\].
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