If in a geometric progression $\left\{ {{a_n}} \right\},\;{a_1} = 3,\;{a_n} = 96$ and ${S_n} = 189$ then the value of $n$ is

Insert two numbers between $3$ and $81$ so that the resulting sequence is $G.P.$

Let $\mathrm{ABC}$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $\mathrm{ABC}$ and the same process is repeated infinitely many times. If $\mathrm{P}$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then:

The sum of the series $5.05 + 1.212 + 0.29088 + ...\,\infty $ is

The sum of first three terms of a $G.P.$ is $\frac{13}{12}$ and their product is $-1$ Find the common ratio and the terms.

Let the first term $a$ and the common ratio $r$ of a geometric progression be positive integers. If the sum of its squares of first three terms is $33033$, then the sum of these three terms is equal to