A) \[\frac{{{2}^{n}}-1}{{{2}^{n\,\,-\,\,1}}}\]
B) \[\frac{{{2}^{n\,-\,1}}-1}{{{2}^{n\,-\,2}}}\]
C) \[2-{{2}^{n}}\]
D) \[\frac{{{2}^{n}}-1}{{{2}^{n}}}\]
Correct Answer: A
Solution :
[a] \[1+\frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+...\,n\] to n terms is a geometric series whose first term a is 1 and the common ratio r is 1/2. \[{{S}_{n}}=\frac{a\,\,(1-{{r}^{n}})}{1-r}=1\frac{\left( 1-\frac{1}{{{2}^{n}}} \right)}{1-\frac{1}{2}}\] \[=\frac{\left( \frac{{{2}^{n}}-1}{{{2}^{n}}} \right)}{\frac{1}{2}}=2.\left( \frac{{{2}^{n}}-1}{{{2}^{n}}} \right)=\frac{{{2}^{n}}-1}{{{2}^{n\,-\,1}}}\]
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