Question

The product of $n$ positive numbers is unity. Then their sum is

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

Hint:

AM $\geq$ GM: Arithmetic mean is always $\geq$ geometric mean, with equality when all numbers are equal.

Answer 1

The Correct Option is D

Step 1: Conceptual Understanding:
Apply the AM--GM inequality to the $n$ positive numbers.
Step 2: Explanation in Detail:
AM $\ge$ GM: $\dfrac{x_1+x_2+\cdots+x_n}{n} \ge (x_1 x_2\cdots x_n)^{1/n} = 1$.
So the sum $\ge n$, i.e.\ it is never less than $n$.
Step 3: Therefore, Stating the Final Answer
The sum is never less than $n$.