If the $A.M.$ and $H.M.$ of two numbers are $27$ and $12$ respectively,then the $G.M.$ of the two numbers will be:

If $a, b, c$ are in $A.P.$ and $a^2, b^2, c^2$ are in $G.P.$ such that $a < b < c$ and $a+b+c = \frac{3}{4}$,then the value of $a$ is

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

Let $b_i > 1$ for $i = 1, 2, \ldots, 101$. Suppose $\log _e b_1, \log _e b_2, \ldots, \log _e b_{101}$ are in Arithmetic Progression $(A.P.)$ with the common difference $\log _e 2$. Suppose $a_1, a_2, \ldots, a_{101}$ are in $A.P.$ such that $a_1 = b_1$ and $a_{51} = b_{51}$. If $t = b_1 + b_2 + \cdots + b_{51}$ and $s = a_1 + a_2 + \cdots + a_{51}$,then:

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

If $a^{1/x} = b^{1/y} = c^{1/z}$ and $a, b, c$ are in $G.P.$,then $x, y, z$ will be in