Question

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 $n^2$

Answer 1

The correct option is D Never less than $n^2$

We know that $A.M.\ge G.M.$

A.M. of $n$ positive numbers is $\frac{1+2+3+...+n}{n}$

G.M. of $n$ positive numbers is $\sqrt[n]{n1\ast 2\ast 3\ast ...\ast n}$

Given that product of $n$ positive numbers is $n^n$

Therefore, G.M. is $\sqrt[n]{n{n}^n}=n$

Hence, $\frac{1+2+3+...+n}{n}\ge n$

$⟹1+2+3+...+n\ge n^2$

Therefore, the sum of n positive numbers is never less than $n^2$