If $a,\;b,\;c$ are in $A.P.$, then $\frac{a}{{bc}},\;\frac{1}{c},\;\frac{2}{b}$ are in

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 ..... .

Let $a$, $b \in R$  be such that $a$, $a + 2b$ , $2a + b$ are in $A.P$. and $(b + 1)^2$, $ab + 5$, $(a + 1)^2$ are in $G.P.$ then $(a + b)$ equals

Suppose $a,\,b,\,c$ are in $A.P.$ and ${a^2},{b^2},{c^2}$ are in $G.P.$ If $a < b < c$ and $a + b + c = \frac{3}{2}$, then the value of $a$ is

If all roots of the equation $x^3 -2ax^2 + 3bx -8$=$0$ are positive, $a$,$b \in R$ , then the minimum value of $b$ is

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