If $G.M. = 18$ and $A.M. = 27$,then $H.M.$ is

Suppose $a, b, c$ are in $A.P.$ and $a^2, b^2, c^2$ are in $G.P.$ If $a < b < c$ and $a + b + c = \frac{3}{2}$,then the value of $a$ is

The $8^{\text{th}}$ common term of the series $S_1 = 3 + 7 + 11 + 15 + 19 + \dots$ and $S_2 = 1 + 6 + 11 + 16 + 21 + \dots$ is $.......$.

What is the minimum value of the sum of the real numbers $a^{-5}, a^{-4}, 3a^{-3}, 1, a^8$,and $a^{10}$ where $a > 0$?

If ${G_1}$ and ${G_2}$ are two geometric means and $A$ is the arithmetic mean inserted between two numbers,then the value of $\frac{{G_1^2}}{{{G_2}}} + \frac{{G_2^2}}{{{G_1}}}$ is

If $\frac{a^{n+1} + b^{n+1}}{a^n + b^n}$ is the $A.M.$ of $a$ and $b$,then $n=$