If the $A.M.$ and $H.M.$ of two numbers are $27$ and $12$ respectively,then the $G.M.$ of the two numbers will be:

In which progression are the terms $\frac{1}{1 + \sqrt{x}}, \frac{1}{1 - x}, \frac{1}{1 - \sqrt{x}}$?

If $a, b, c$ are positive integers,then $(a + b)(b + c)(c + a)$ is

If $a, x, y, z, b$ are in Arithmetic Progression ($A$.$P$.) such that $x + y + z = 15$,and if $a, x, y, z, b$ are in Harmonic Progression ($H$.$P$.) such that $1/x + 1/y + 1/z = 5/3$,find the values of $a$ and $b$.

If $a, b, c$ are in $A.P.$,then $10^{ax + 10}, 10^{bx + 10}, 10^{cx + 10}$ will be in

The ratio of the $A.M.$ and $G.M.$ of two positive numbers $a$ and $b$ is $m: n.$ Show that $a: b = (m + \sqrt{m^{2} - n^{2}}) : (m - \sqrt{m^{2} - n^{2}}).$