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  • The least positive integer n such that <img alt="1-\frac{2}{3}-\frac{2}{3^{2}}-............-\frac{2}{3^{n-1}}< \frac{1}{100}" src="https://learn.careers360.com/latex-image/?%5Cdpi%7B100%7D%201-%

The least positive integer n such that 1-\frac{2}{3}-\frac{2}{3^{2}}-............-\frac{2}{3^{n-1}}< \frac{1}{100}  is

Option: 1

4


Option: 2

5


Option: 3

6


Option: 4

7


Answers (1)

best_answer

Use @4 

1 - \frac{2}{3}- \frac{2}{3^{2}}- \cdot \cdot \cdot - \frac{2}{3n-1}< \frac{1}{100}

Let S= 1 - \frac{2}{3} - \frac{2}{3^{2}}\cdot \cdot -\frac{2}{3^{n-1}}

\frac{\frac{S}{3}}{S-\frac{S}{3}}= \frac{\frac{1}{3}-\frac{2}{3^{2}}- \frac{2}{3^{3}}\cdot \cdot \cdot - \frac{2}{3^{n-1}}- \frac{2}{3^{n}}}{1 - \frac{1}{3}-\frac{2}{3}\cdot \cdot \cdot - \frac{2}{3^{n}}}

\frac{2S}{3} = \frac{2}{3^{n}}

\therefore S = \frac{1}{3^{n-1}}

\therefore \frac{1}{3^{n-1}} < \frac{1}{100}

\therefore 3^{n-1}> 100

SD minimum value of n is 6

 

Posted by

Gautam harsolia

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