Question

The product of $n$ positive numbers is unity. Their sum is

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

Answer 1

The correct option is D

never less than $n$

Explanation for the correct option:

Let the numbers be $a_1,a_2,.........,a_n$

Given: $a_1·a_2·.........,a_n=1$

As we know that $AM\ge GM$

$$\begin{gathered} AM=\frac{a_1+a_2+a_3+..........+a_n}{n} \\ GM=\sqrt[n]{a_1·a_2·...........·a_n} \end{gathered}$$

So,

$$\begin{gathered} \frac{a_1+a_2+a_3+..........a_n}{n}\ge \sqrt[n]{a_1·a_{2............}{a}_n} \\ \Rightarrow \frac{a_1+a_2+a_3+..........a_n}{n}\ge \sqrt[n]{1} \\ \Rightarrow \frac{a_1+a_2+a_3+..........a_n}{n}\ge 1 \\ \Rightarrow a_1+a_2+a_3+..........a_n\ge n \end{gathered}$$

Therefore, the required sum is never less than $n$.

Hence, option D is the correct answer.