The product of $n$ positive numbers is $1$. The sum of these numbers cannot be less than what value?

If ${G_1}$ and ${G_2}$ are two geometric means and $A$ is the arithmetic mean inserted between two numbers,then the value of $\frac{{G_1^2}}{{{G_2}}} + \frac{{G_2^2}}{{{G_1}}}$ is

Let $G$ be the geometric mean of two positive numbers $a$ and $b,$ and $M$ be the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}.$ If $\frac{1}{M} : G$ is $4:5,$ then $a:b$ can be

The sum of three numbers in $G.P.$ is $56$. If we subtract $1, 7, 21$ from these numbers in that order,we obtain an arithmetic progression. Find the numbers.

If the altitudes of a triangle are in $A.P.$,then the sides of the triangle are in

If $a, b, c$ are in $A.P.$ and $a^2, b^2, c^2$ are in $H.P.$,then