Suppose $a, b, c$ are in $A.P.$ and $a^{2}, 2b^{2}, c^{2}$ are in $G.P.$ If $a < b < c$ and $a+b+c=1,$ then $9(a^{2}+b^{2}+c^{2})$ is equal to . . . . . . .

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

If $x = \sum_{n=0}^{\infty} a^{n}$,$y = \sum_{n=0}^{\infty} b^{n}$,$z = \sum_{n=0}^{\infty} c^{n}$,where $a, b, c$ are in $A.P.$ and $|a| < 1, |b| < 1, |c| < 1$,$abc \neq 0$,then:

If the ratio of two numbers is $9:1$,then the ratio of their geometric mean to their harmonic mean is

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an Arithmetic Progression are equal to the terms of a Geometric Progression,and these terms are $x, y, z$ respectively,then $x^{y - z} \cdot y^{z - x} \cdot z^{x - y} = \dots$

If $a^{1/x} = b^{1/y} = c^{1/z}$ and $a, b, c$ are in geometric progression,then $x, y, z$ are in.....