The product of $n$ positive numbers is unity. Their sum is

$A$ car completes the first half of its journey with a velocity $v_1$ and the rest half with a velocity $v_2$. Then the average velocity of the car for the whole journey is

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

If the arithmetic mean of two numbers $a$ and $b$ is twice their geometric mean,then $a : b = \dots$

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 $a, b, c$ are in geometric progression,then in which progression are $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ such that the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have common roots?