Let $a, b$ and $c$ be the $7^{th},\,11^{th}$ and $13^{th}$ terms respectively of a non -constant $A.P.$ If these are also the three consecutive terms of a $G.P.$ then $\frac {a}{c}$ is equal to

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 $a,\,b,\,c,\,d$ are positive real numbers such that $a + b + c + d$ $ = 2,$ then $M = (a + b)(c + d)$ satisfies the relation

If $\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}(x \ne 0)$, then $a,\;b,\;c,\;d$ are in

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 $a,\,b,\,c$ 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