If $a,\;b,\;c$ are the positive integers, then $(a + b)(b + c)(c + a)$ is

If the ${p^{th}},\;{q^{th}}$ and ${r^{th}}$ term of a $G.P.$ and $H.P.$ are $a,\;b,\;c$, then $a(b - c)\log a + b(c - a)$ $\log b + c(a - b)\log c = $

If the arithmetic mean and geometric mean of the $p ^{\text {th }}$ and $q ^{\text {th }}$ terms of the sequence $-16,8,-4,2, \ldots$ satisfy the equation $4 x^{2}-9 x+5=0,$ then $p+q$ is equal to ..... .

The product of $n$ positive numbers is unity. Their sum is

If $a, b, c$ are in $G.P.$ and $q^{\frac{1}{x}}=k^{\frac{1}{y}}=c^{\frac{1}{2}},$ prove that $x, y, z$ are in $A.P.$

Let ${a_1},\;{a_2},.........{a_{10}}$ be in $A.P.$ and ${h_1},\;{h_2},........{h_{10}}$ be in $H.P.$ If ${a_1} = {h_1} = 2$ and ${a_{10}} = {h_{10}} = 3$, then  ${a_4}{h_7}$ is