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 the geometric mean between $a$ and $b$ is $\frac{a^{n+1} + b^{n+1}}{a^n + b^n}$,then what is the value of $n$?

If $a, b, c$ are in $A.P.$ and $|a|, |b|, |c| < 1$,and $x = 1 + a + a^2 + \dots \infty$,$y = 1 + b + b^2 + \dots \infty$,$z = 1 + c + c^2 + \dots \infty$,then $x, y, z$ shall be in:

If $9$ arithmetic means $(A.M.s)$ and $9$ harmonic means $(H.M.s)$ are inserted between $2$ and $3$,and if the harmonic mean $H$ corresponds to the arithmetic mean $A$ (i.e.,the $j^{th}$ $A.M.$ and $j^{th}$ $H.M.$),then $A + \frac{6}{H} = $

Let $\frac{1}{16}, a$ and $b$ be in $G.P.$ and $\frac{1}{a}, \frac{1}{b}, 6$ be in $A.P.,$ where $a, b > 0$. Then $72(a + b)$ is equal to ...... .

If the arithmetic mean between $p$ and $q$ $(p > q)$ is twice the geometric mean,then $p : q = .......$