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

If $\alpha, \beta$ are the roots of $x^2 - 3x + a = 0$ and $\gamma, \delta$ are the roots of $x^2 - 12x + b = 0$,and the numbers $\alpha, \beta, \gamma, \delta$ (in order) form an increasing $G.P.$,then:

The sum of $3$ numbers in geometric progression is $38$ and their product is $1728$. The middle number is

Consider an infinite geometric series with first term $a$ and common ratio $r$. If its sum is $4$ and the second term is $3/4$,find the values of $a$ and $r$.

Let $A_{1}, A_{2}, A_{3}, \ldots$ be squares such that for each $n \geq 1,$ the length of the side of $A_{n}$ equals the length of the diagonal of $A_{n+1}$. If the side length of $A_{1}$ is $12 \text{ cm}$,then the smallest value of $n$ for which the area of $A_{n}$ is less than $1 \text{ cm}^2$ is:

The sum of an infinite geometric progression is $\frac{4}{3}$ and the first term is $\frac{3}{4}$. The common ratio is