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 $a, b, c > 1, a^3, b^3$ and $c^3$ be in $A.P.$, and $\log _a b$, $\log _c a$ and $\log _b c$ be in G.P. If the sum of first $20$ terms of an $A.P.$, whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-444$, then abc is equal to

The common difference of an $A.P.$ whose first term is unity and whose second, tenth and thirty fourth terms are in $G.P.$, is

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 $A$ and $G$ are arithmetic and geometric means and ${x^2} - 2Ax + {G^2} = 0$, then

If ${A_1},\;{A_2};{G_1},\;{G_2}$ and ${H_1},\;{H_2}$ be $AM's,\;GM's$ and $HM's$ between two quantities, then the value of $\frac{{{G_1}{G_2}}}{{{H_1}{H_2}}}$ is