If $a^2(b + c), b^2(c + a), c^2(a + b)$ are in Arithmetic Progression $(AP)$,then $a, b, c$ are in which progression?

Three non-zero real numbers form an $A.P.$ and the squares of these numbers taken in the same order form a $G.P.$ Then the number of all possible common ratios of the $G.P.$ is

Given an $A.P.$ and a $G.P.$ with positive terms, where the first and second terms of both progressions are equal. If $a_n$ and $b_n$ are the $n^{\text{th}}$ terms of the $A.P.$ and $G.P.$ respectively, then:

If $a, b, c$ are in $AP$,and $b-a, c-b, a$ are in $GP$,then $a: b: c$ is

Let $G$ be the geometric mean of two positive numbers $a$ and $b,$ and $M$ be the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}.$ If $\frac{1}{M} : G$ is $4:5,$ then $a:b$ can be

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 . . . . . . .