If five $G.M.s$ are inserted between $486$ and $2/3$,then the fourth $G.M.$ will be:

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

The third term of a $G.P.$ is the square of the first term. If the second term is $8$,then the $6^{th}$ term is:

Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b+2$,then the value of $\frac{a^2+a-14}{a+1}$ is

The sum of three numbers in a Geometric Progression $(GP)$ is $38$ and their product is $1728$. The largest of these numbers is.......

Let $S_1$ be the sum of areas of the squares whose sides are parallel to the coordinate axes. Let $S_2$ be the sum of areas of the slanted squares as shown in the figure. Then,$\frac{S_1}{S_2}$ is equal to