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

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

If $x, 1, z$ are in $A.P.$ and $x, 2, z$ are in $G.P.$,then $x, 4, z$ will be in

Let $a_1, a_2, a_3$ be any positive real numbers,then which of the following statements is not true?

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

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} = $