A) \[\frac{3({{3}^{n}}-1)}{4}-1\]
B) \[\frac{3({{3}^{n}}-1)-2n}{4}\]
C) \[\frac{3({{3}^{n}}-1)-n}{4}\]
D) \[\frac{2n-3({{3}^{n}}-1)}{4}\]
E) \[\frac{3({{3}^{n}}-1)-n}{2}\]
Correct Answer: B
Solution :
\[{{T}_{n}}=1+3+9+27+...n\,\,term\] \[=1+3+{{3}^{2}}+{{3}^{3}}+...n\,\,term\] \[=1.\frac{({{3}^{n}}-1)}{3-1}=\frac{{{3}^{n}}-1}{2}\] \[\therefore \] \[S=\Sigma {{T}_{n}}=\frac{1}{2}\Sigma {{3}^{n}}-\frac{1}{2}\Sigma 1\] \[=\frac{1}{2}[3+{{3}^{2}}+{{3}^{3}}+....+{{3}^{n}}]-\frac{n}{2}\] \[=\frac{1}{2}\left[ \frac{3.({{3}^{n}}-1)}{3-1} \right]-\frac{n}{2}=\frac{3({{3}^{n}}-1)-2n}{4}\]
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