Question

The product of \(n\) positive numbers is unity, then their sum is:

• A. a positive integer
• B. divisible by \(n\)
• C. equal to \(n+\frac{1}{n}\)
• D. never less than \(n\)

Hint:

AM–GM inequality gives minimum sum for fixed product.

Answer 1

The Correct Option is D

Step 1: By AM–GM inequality: \[ \frac{x_1+x_2+\cdots+x_n}{n}\ge (x_1x_2\cdots x_n)^{1/n} \]
Step 2: Given product \(=1\): \[ \text{Sum}\ge n \]