Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic progression. If $S_{3n} = 3S_{2n}$,then the value of $\frac{S_{4n}}{S_{2n}}$ is:

If the roots of the equation $4x^3 - 12x^2 + 11x + k = 0$ are in arithmetic progression,then $k$ is equal to

If $x, y, z$ are three real numbers with the same sign,then the value of $\frac{x}{y} + \frac{y}{z} + \frac{z}{x}$ lies in which interval?

If the sum of $n$ terms of the sequence $2, 5, 8, 11, \dots$ is $60100$,then $n = \dots$

For $p, q \in R$,consider the real-valued function $f(x) = (x - p)^2 - q$,where $x \in R$ and $q > 0$. Let $a_1, a_2, a_3, a_4$ be in an arithmetic progression with mean $p$ and a positive common difference $d$. If $|f(a_i)| = 500$ for all $i = 1, 2, 3, 4$,then the absolute difference between the roots of $f(x) = 0$ is:

If $a, b, c$ are in $A.P.$,then $3^a, 3^b, 3^c$ shall be in