A) \[2-\frac{3}{{{2}^{17}}}\]
B) \[2-\frac{11}{{{2}^{19}}}\]
C) \[1-\frac{11}{{{2}^{20}}}\]
D) \[2-\frac{21}{{{2}^{20}}}\]
Correct Answer: B
Solution :
\[S=\sum\limits_{k=1}^{20}{\frac{1}{{{2}^{k}}}}\] \[S=\frac{1}{2}+\frac{2}{{{2}^{2}}}+\frac{3}{{{3}^{2}}}+...+\frac{20}{{{2}^{20}}}\] \[S\times \frac{1}{2}=\frac{1}{{{2}^{2}}}+\frac{2}{{{2}^{3}}}+...+\frac{19}{{{2}^{20}}}+\frac{20}{{{2}^{21}}}\] \[\Rightarrow \]\[\left( 1-\frac{1}{2} \right)S=\frac{1}{2}+\frac{1}{{{2}^{2}}}+...+\frac{1}{{{2}^{20}}}-\frac{20}{{{2}^{21}}}\] \[\Rightarrow \]\[S=2-\frac{11}{{{2}^{19}}}\]
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