If $a, b, c$ are any three positive numbers,then what is the minimum value of $(a + b + c) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)$?

If $a$ and $b$ are two different positive real numbers,then which of the following relations is true?

If the $2^{nd}, 5^{th},$ and $9^{th}$ terms of a non-constant $A.P.$ are in $G.P.$,then the common ratio of this $G.P.$ is:

If $A$ and $G$ are the $A.M.$ and $G.M.$ respectively between two positive numbers,prove that the numbers are $A \pm \sqrt{(A + G)(A - G)}$.

Let the first three terms $2, p$ and $q$,with $q \neq 2$,of a $G.P.$ be respectively the $7^{\text{th}}$,$8^{\text{th}}$ and $13^{\text{th}}$ terms of an $A.P.$ If the $5^{\text{th}}$ term of the $G.P.$ is the $n^{\text{th}}$ term of the $A.P.$,then $n$ is equal to

The arithmetic mean and the geometric mean of two distinct $2$-digit numbers $x$ and $y$ are two integers,one of which can be obtained by reversing the digits of the other (in base $10$ representation). Then,$x+y$ equals