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

If the $A.M., G.M.$ and $H.M.$ between two positive numbers $a$ and $b$ are equal, then

If three unequal numbers $p,\;q,\;r$ are in $H.P.$ and their squares are in $A.P.$, then the ratio $p:q:r$ is

Let $x, y, z$ be three non-negative integers such that $x+y+z=10$. The maximum possible value of $x y z+x y+y z+z x$ is

If each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1=\frac{1}{8}$ and $a_2 \neq a_1$, is the arithmetic mean of the next two terms and $S_n=a_1+a_2+\ldots+a_n$, then $\mathrm{S}_{20}-\mathrm{S}_{18}$ is equal to

The $A.M., H.M.$ and $G.M.$ between two numbers are $\frac{{144}}{{15}}$, $15$ and $12$, but not necessarily in this order. Then $H.M., G.M.$ and $A.M.$ respectively are