When $\frac{1}{a} + \frac{1}{c} + \frac{1}{{a - b}} + \frac{1}{{c - d}} = 0$ and $b \ne a \ne c$, then $a,\;b,\;c$ are

If three unequal non-zero real numbers $a,\;b,\;c$ are in $G.P.$ and $b - c,\;c - a,\;a - b$ are in $H.P.$, then the value of $a + b + c$ is independent of

Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b+2$, then the value of $\frac{a^2+a-14}{a+1}$ is

Let $a, b, c, d\, \in \, R^+$ and $256\, abcd \geq  (a+b+c+d)^4$ and $3a + b + 2c + 5d = 11$ then $a^3 + b + c^2 + 5d$ 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 the $A.M.$ and $H.M.$ of two numbers is $27$ and $12$ respectively, then $G.M.$ of the two numbers will be