Question

If the product of $n$ positive real numbers is one, then their sum is

• A. $n+\frac{1}{n}$
• B. $n-\frac{1}{n}$
• C. $2n+\frac{1}{n}$
• D. never less than n

Answer 1

Answer: (D)

never less than n

Answer 2

The correct option is(D): never less than n.

Let ${{x}_{1}},{{x}_{2}},.....{{x}_{n}}$ are n positive integers.
$\because$ $AM\ge GM$
$\therefore$ $\frac{{{x}_{1}}+{{x}_{2}}+....+{{x}_{n}}}{n}\ge {{({{x}_{1}}.{{x}_{2}}....{{x}_{n}})}^{1/n}}$
$\Rightarrow$ $\frac{{{x}_{1}}+{{x}_{2}}+....+{{x}_{n}}}{n}\ge \,\,{{(1)}^{1/n}}$
$\Rightarrow$ ${{x}_{1}}+{{x}_{2}}+...+{{x}_{n}}\ge n$
Hence, their sun is never less than n.