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  4. The least positive integer n such that 1-(2/3)-(2/32)-.......-(2/3n-1) < (1/100), is :

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

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Solution:

$1-2\left(\frac{1}{3^{1}}+\frac{1}{3^{2}} \ldots \ldots . \frac{1}{3^{ n -1}}\right)<\frac{1}{100}$
$1-2\left(\frac{1}{3}\right) \frac{\left[1-\frac{1}{3^{ n -1}}\right]}{\left(\frac{2}{3}\right)}<\frac{1}{100}$
$\Rightarrow 1-1+\frac{1}{3^{ n -1}}<\frac{1}{100}$
$100<3^{ n -1}$
$n -1=5$
$n =6$