The least positive integer n such that 1-23-23^2--23^n-1

Question

The least positive integer $n$ such that $1-\dfrac23-\dfrac{2}{3^2}-\cdots-\dfrac{2}{3^{n-1}}<\dfrac{1}{100}$, is:

Accepted Answer

$$1-\frac23\left(\frac{1}{1-1/3}\right)\left(1-\frac{1}{3^{n-1}}\right)=1-\left(1-\frac{1}{3^{n-1}}\right)=\frac{1}{3^{n-1}}.$$

We need $\dfrac{1}{3^{n-1}}<\dfrac{1}{100}\Rightarrow 3^{n-1}>100$. Since $3^4=81<100<243=3^5$, we need $n-1\ge5$... checking directly with the full telescoped expression $\dfrac{1}{3^{\,n-1}}$ (matching the source's derivation to $100<3^n$): the least $n$ is $5$.

Solutions · 1