If the $2^{nd}\,, \,5^{th}\,\, and \,\,9^{th}$ terms of a non-constant $A.P.$ are in $G.P.$, then the common ratio of this $G.P.$ is :

If the $A.M.$ is twice the $G.M.$ of the numbers $a$ and $b$, then $a:b$ will be

If $a, b, c$ are in $GP$ and $4a, 5b, 4c$ are in $AP$ such that $a + b + c = 70$,  then value of $a^3 + b^3 + c^3$ is

If $a,\,b,\,c$ are three unequal numbers such that $a,\,b,\,c$ are in $A.P.$ and $b -a, c -b, a$ are in $G.P.$, then $a : b : c$ is

Let $n \geq 3$ be an integer. For a permutation $\sigma=\left(a_1, a_2, \ldots, a_n\right)$ of $(1,2, \ldots, n)$ we let $f_\sigma(x)=a_n x^{n-1}+a_{n-1} x^{n-2}+\ldots a_2 x+a_1$. Let $S_\sigma$ be the sum of the roots of $f_\sigma(x)=0$ and let $S$ denote the sum over all permutations $\sigma$ of $(1,2, \ldots, n)$ of the numbers $S_\sigma$. Then,

If $a,\;b,\;c$ are in $A.P.$ and $a,\;c - b,\;b - a$ are in $G.P. $ $(a \ne b \ne c)$, then $a:b:c$ is