Q. If the product of $n$ positive numbers is $n^{n}$, then their sum is
A) A positive integer
B) Divisible by $n$
C) Equal to $n + \frac{1}{n}$
D) Never less than $n^{2}$
Explanation
(d)
Let $a_{1},a_{2},\ldots,a_{n}\$be $n$ positive integers such that
$a_{1}a_{2}\ldots a_{n} = n^{n}$
Since, $AM \geq GM$
$\therefore\frac{a_{1} + a_{2} + ... + a_{n}}{n} \geq \left( a_{1}a_{2}\ldots a_{n} \right)^{1/n}$
$\Rightarrow \frac{a_{1} + a_{2} + ... + a_{n}}{n} \geq n$
$\Rightarrow a_{1} + a_{2} + ... + a_{n} \geq n^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.