Let ${a_1, a_2, \dots, a_{10}}$ be in $A.P.$ and ${h_1, h_2, \dots, h_{10}}$ be in $H.P.$ If ${a_1 = h_1 = 2}$ and ${a_{10} = h_{10} = 3}$,then the value of ${a_4 h_7}$ is:

Let three real numbers $a, b, c$ be in arithmetic progression and $a+1, b, c+3$ be in geometric progression. If $a > 10$ and the arithmetic mean of $a, b$ and $c$ is $8$,then the cube of the geometric mean of $a, b$ and $c$ is

If ${A_1}, {A_2}$; ${G_1}, {G_2}$ and ${H_1}, {H_2}$ are $AM's$,$GM's$ and $HM's$ between two quantities,then the value of $\frac{{{G_1}{G_2}}}{{{H_1}{H_2}}}$ is

Let $E = x^{2017} + y^{2017} + z^{2017} - 2017xyz$ (where $x, y, z \geq 0$),then the least value of $E$ is

The value of $n$ for which $\frac{x^{n+1}+y^{n+1}}{x^{n}+y^{n}}$ is the geometric mean of $x$ and $y$ is

If $\alpha, \beta, \gamma$ are the geometric means between $ca, ab$; $ab, bc$; and $bc, ca$ respectively,where $a, b, c$ are in $A.P.$,then $\alpha^2, \beta^2, \gamma^2$ are in