If the ${(m + 1)^{th}},\;{(n + 1)^{th}}$ and ${(r + 1)^{th}}$ terms of an $A.P.$ are in $G.P.$ and $m,\;n,\;r$ are in $H.P.$, then the value of the ratio of the common difference to the first term of the $A.P.$ is

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 the first and ${(2n - 1)^{th}}$ terms of an $A.P., G.P.$ and $H.P.$ are equal and their ${n^{th}}$ terms are respectively $a,\;b$ and $c$, then

Let $A, G$ and $H$ be the arithmetic mean, geometric mean and harmonic mean, respectively of two distinct positive real numbers. If $\alpha$ is the smallest of the two roots of the equation $A(G-H) x^2+G(H-A) x$ $+H(A-G)=0$ then,

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. . . . . . 

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