If ${\log _x}y,\;{\log _z}x,\;{\log _y}z$ are in $G.P.$ $xyz = 64$ and ${x^3},\;{y^3},\;{z^3}$ are in $A.P.$, then

If $a,\,b,\,c,\,d$ are positive real numbers such that $a + b + c + d$ $ = 2,$ then $M = (a + b)(c + d)$ satisfies the relation

If the arithmetic and geometric means of $a$ and $b$ be $A$ and $G$ respectively, then the value of $A - G$ will be

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 $2^{\text {nd }}, 8^{\text {th }}$ and $44^{\text {th }}$, terms of a non-constant $A.P.$ be respectively the $1^{\text {st }}, 2^{\text {nd }}$ and $3^{\text {rd }}$ terms of $G.P.$ If the first term of $A.P.$ is $1$ then the sum of first $20$ terms is equal to-

Let $3, a, b, c$ be in $A.P.$ and $3, a-1, b+1, c+9$ be in $G.P.$ Then, the arithmetic mean of $a, b$ and $c$ is :