Q. The product of $n$ positive numbers is unity. Their sum is
A) A positive integer
B) Equal to $n + \frac{1}{n}$
C) Divisible by $n$
D) Never less than $n$
Explanation
(d)
Given that, $x_{1}x_{2}x_{3}\ldots x_{n} = 1$ …(i)
We know that, AM $\geq$ GM
$\therefore\ \ \left( \frac{x_{1} + x_{2} + x_{3} + ... + x_{n}}{n} \right) \geq \left( x_{1}x_{2}x_{3}\ldots x_{n} \right)^{1/n}$
$= (1)^{1/n} = 1$ [from Eq.(i)]
$\Rightarrow \ x_{1} + x_{2} + x_{3} + ... + x_{n} \geq n$
$\therefore\ x_{1} + x_{2} + x_{3} + ... + x_{n}$
Can never be less than $n$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.