If the roots of the cubic equation $ax^3 + bx^2 + cx + d = 0$ are in $G.P.$,then

Let $A_n = \left( \frac{3}{4} \right) - \left( \frac{3}{4} \right)^2 + \left( \frac{3}{4} \right)^3 - \dots + (-1)^{n-1} \left( \frac{3}{4} \right)^n$ and $B_n = 1 - A_n$. Then,the least odd natural number $p$ such that $B_n > A_n$ for all $n \geq p$ is:

Find the Geometric Mean $(G.M.)$ of the sequence $1, 2, 2^2, \dots, 2^n$.

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:

Let $f(x)$ be a function such that $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in \mathbb{N}$. If $f(1)=3$ and $\sum_{k=1}^{n} f(k)=3279$,then the value of $n$ is:

If $x, y, z$ are in geometric progression and $a^x = b^y = c^z$,then . . . . . .