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

If $s$ is the sum of an infinite $G.P.$ and $a$ is the first term,then the common ratio $r$ is given by:

If the $4^{\text{th}}$,$10^{\text{th}}$,and $16^{\text{th}}$ terms of a $G.P.$ are $x, y$,and $z$ respectively,prove that $x, y, z$ are in $G.P.$

If the $5^{th}$ term of a $G.P.$ is $\frac{1}{3}$ and the $9^{th}$ term is $\frac{16}{243}$,then the $4^{th}$ term will be:

The product of three consecutive terms of a geometric progression is $216$ and the sum of the products of these terms taken two at a time is $156$. Find the terms.

$A$ person has $2$ parents,$4$ grandparents,$8$ great-grandparents,and so on. Find the total number of his ancestors during the $10$ generations preceding his own.