A) \[2n-\frac{4}{3}\left\{ 1-\frac{1}{{{2}^{2n}}} \right\}\]
B) \[2n+\frac{4}{3}\left\{ 1-\frac{1}{{{2}^{2n}}} \right\}\]
C) \[2n+\frac{4}{3}\left\{ 1+\frac{1}{{{2}^{2n}}} \right\}\]
D) None of these
Correct Answer: A
Solution :
| \[{{S}_{n}}=1+\frac{1-{{\left( \frac{1}{2} \right)}^{3}}}{1-\frac{1}{2}}+\frac{1-{{\left( \frac{1}{2} \right)}^{5}}}{1-\frac{1}{2}}+....n\] terms \[=1+2\left( 1-{{\left( \frac{1}{2} \right)}^{3}} \right)+2\left( 1-{{\left( \frac{1}{2} \right)}^{5}} \right)+....n\] |
| \[times=1+(2+2+.....(n-1)\,\,times)\]\[-\,\,\left[ {{\left( \frac{1}{2} \right)}^{2}}+{{\left( \frac{1}{2} \right)}^{4}}+......(n-1)\,\,times \right]\] |
| \[=1+2\,(n+1)-\frac{{{(1/2)}^{2}}}{1-{{\left( \frac{1}{2} \right)}^{2}}}\left( 1-{{\left( \frac{1}{2} \right)}^{2\,(n-\,1)}} \right)\] |
| \[=2n-1-\frac{1}{3}+\frac{4}{{{3.2}^{2n}}}=2n-\frac{4}{3}\left( 1-\frac{1}{{{2}^{2n}}} \right)\] |
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