A) 4
B) 5
C) 7
D) 7
Correct Answer: B
Solution :
\[1-\frac{2}{3}-\frac{2}{{{3}^{2}}}....\frac{2}{{{3}^{n-1}}}<\frac{1}{100}\] \[\Rightarrow \]\[1-\frac{2}{3}\left[ \frac{1}{3}+\frac{1}{{{3}^{2}}}+\frac{1}{{{3}^{3}}}+...\frac{1}{{{3}^{n-1}}} \right]<\frac{1}{100}\] \[\Rightarrow \]\[\frac{1-2\left[ \frac{1}{3}\left( \frac{1}{{{3}^{n}}}-1 \right) \right]}{\frac{1}{3}-1}<\frac{1}{100}\] \[\Rightarrow \]\[1-2\left[ \frac{{{3}^{n}}-1}{{{2.3}^{n}}} \right]<\frac{1}{100}\]\[\Rightarrow \]\[1-\left[ \frac{{{3}^{n}}-1}{{{2.3}^{n}}} \right]<\frac{1}{100}\] \[\Rightarrow \]\[1-1+\frac{1}{{{3}^{n}}}<\frac{1}{100}\]\[\Rightarrow \]\[100<{{3}^{n}}\]
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