The sum of the first five terms of the series $3 + 4\frac{1}{2} + 6\frac{3}{4} + \dots$ will be

If the sum of the first $6$ terms is $9$ times the sum of the first $3$ terms of the same $G.P.$,then the common ratio of the series will be

For a $G.P.$,if $(m+n)^{\text{th}}$ term is $p$ and $(m-n)^{\text{th}}$ term is $q$,then the $m^{\text{th}}$ term is $.........$

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:

$0.14189189189...$ can be expressed as a rational number.

In a geometric progression,the first term is $1$. If $4T_2 + 5T_3$ is minimum,then what is the common ratio?