If ${A_1},\;{A_2}$ are the two $A.M.'s$ between two numbers $a$ and $b$ and ${G_1},\;{G_2}$ be two $G.M.'s$ between same two numbers, then $\frac{{{A_1} + {A_2}}}{{{G_1}.{G_2}}} = $

If ${A_1},\;{A_2};{G_1},\;{G_2}$ and ${H_1},\;{H_2}$ be two $A.M.s$, $G.M.s$ and $H.M.s$ between two numbers respectively, then $\frac{{{G_1}{G_2}}}{{{H_1}{H_2}}} \times \frac{{{H_1} + {H_2}}}{{{A_1} + {A_2}}}$ =

Let $x, y, z$  be positive real numbers such that $x + y + z = 12$ and  $x^3y^4z^5 = (0. 1 ) (600)^3$. Then $x^3 + y^3 + z^3$ is equal to

Let $m$ be the minimum possible value of $\log _3\left(3^{y_1}+3^{y_2}+3^{y_3}\right)$, where $y _1, y _2, y _3$ are real numbers for which $y _1+ y _2+ y _3=9$. Let $M$ be the maximum possible value of $\left(\log _3 x _1+\log _3 x _2+\log _3 x _3\right)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1+x_2+x_3=9$. Then the value of $\log _2\left(m^3\right)+\log _3\left(M^2\right)$ is. . . . . . 

Given a sequence of $4$ numbers, first three of which are in $G.P.$ and the last three are in $A.P$. with common difference six. If first and last terms in this sequence are equal, then the last term is

If the ratio of $H.M.$ and $G.M.$ between two numbers $a$ and $b$ is $4:5$, then the ratio of the two numbers will be