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 ..... .

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

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

Let $a , b , c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5 b^3 c^2 d$ is $3750 \beta$, then the value of $\beta$ is

If the $A.M.$ of two numbers is greater than $G.M.$ of the numbers by $2$ and the ratio of the numbers is $4:1$, then the numbers are

The ratio of the $A.M.$ and $G.M.$ of two positive numbers $a$ and $b,$ is $m: n .$ Show that $a: b=(m+\sqrt{m^{2}-n^{2}}):(m-\sqrt{m^{2}-n^{2}})$