If the arithmetic, geometric and harmonic means between two distinct positive real numbers be $A,\;G$ and $H$ respectively, then the relation between them is

If $a,\,b,\;c$ are in $A.P.$ and ${a^2},\;{b^2},\;{c^2}$ are in $H.P.$, then

Three numbers are in an increasing geometric progression with common ratio $\mathrm{r}$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $\mathrm{d}$. If the fourth term of GP is $3 \mathrm{r}^{2}$, then $\mathrm{r}^{2}-\mathrm{d}$ is equal to:

Let $a, b$ and $c$ be in $G.P$ with common ratio $r,$ where $a \ne 0$ and $0\, < \,r\, \le \,\frac{1}{2}$.  If $3a, 7b$ and $15c$ are the first three terms of an $A.P.,$ then the $4^{th}$ term of this $A.P$ 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,

$a,\,g,\,h$ are arithmetic mean, geometric mean and harmonic mean between two positive numbers $x$ and $y$ respectively. Then identify the correct statement among the following