The least positive integer n such that 1−23−232−⋯−23n−1<1100 is

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

Solution:

$\begin{array}{l} \frac{99}{100} < \frac{2}{3} + \frac{2}{3^{2}} + \dots + \frac{2}{3^{n - 1}} \\ \Rightarrow \frac{(2 / 3) [1 - (1 / 3)^{n - 1}]}{1 - 1 / 3} > \frac{99}{100} \\ \Rightarrow \frac{1}{100} > \frac{1}{3^{n - 1}} \Rightarrow 3^{n - 1} > 100 \\ \Rightarrow n - 1 > 5 \Rightarrow n > 6 \end{array}$

Thus, least value of $n$ is 7.

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