A) 2
B) 6
C) 5
D) none of these
Correct Answer: B
Solution :
Let \[S=1+\frac{3}{2}+\frac{5}{{{2}^{2}}}+\frac{7}{{{2}^{3}}}+...\] Given series is an AGP whose common ratio is \[\frac{1}{2}\]. \[S=1\frac{3}{2}+\frac{5}{{{2}^{2}}}+\frac{7}{{{2}^{3}}}+....\] \[\frac{1}{2}S=\frac{1}{2}+\frac{3}{{{2}^{2}}}+\frac{5}{{{2}^{3}}}+...\] ----------------------------------------------- \[\frac{1}{2}S=1+\frac{2}{2}+\frac{2}{{{2}^{2}}}+\frac{2}{{{2}^{3}}}+...\] \[=1+2\left( \frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+... \right)\] \[=1+2\left( \frac{1/2}{1-1/2} \right)\] \[=1+2=3\] \[\Rightarrow \] \[S=6\]
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