A) \[2(n-1)+\frac{1}{{{2}^{n-1}}}\]
B) \[2n-\frac{1}{{{2}^{n}}}\]
C) \[2+\frac{1}{{{2}^{n}}}\]
D) \[2n-1+\frac{1}{{{2}^{n}}}\]
Correct Answer: A
Solution :
Let \[S=1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+...\] \[=1+\frac{(4-1)}{2}+\frac{(8-1)}{4}+\frac{(16-1)}{8}+\frac{(32-1)}{16}+...\] \[=1+2-\frac{1}{2}+2-\frac{1}{4}+2-\frac{1}{8}+2-\frac{1}{16}+...\] \[=1+2(n-1)-\left[ \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...(n-1) \right]\] \[=1+2(n-1)-\left[ \frac{\frac{1}{2}\left( 1-\frac{1}{{{2}^{n-1}}} \right)}{1-\frac{1}{2}} \right]\] \[=1+2(n-1)-1+\frac{1}{{{2}^{n-1}}}\] \[=2(n-1)+\frac{1}{{{2}^{n-1}}}\]
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