Question

If $a_1,a_2,....a_n$ are positive numbers such that $a_1·a_2·a_3·....·a_n=1$, then their sum is:

• A. A positive integer.
• B. Divisible by $n$.
• C. Never less than $n$.
• D. None of these.

Answer 1

The correct option is C

Never less than $n$.

Explanation for the correct option:

Finding the sum of ${\mathit{a}}_{\mathbf{1}}\mathbf{,}\mathbf{}{\mathit{a}}_{\mathbf{2}}\mathbf{,}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{}{\mathit{a}}_{\mathbf{n}}$:

We know that the Arithmetic Mean of positive numbers is $\ge$ their Geometric Mean.

As arithmetic mean $=\frac{\mathrm{Sum}\mathrm{of}\mathrm{all}\mathrm{observations}}{\mathrm{No}.\mathrm{of}\mathrm{observations}}$ and geometric mean $=\sqrt[n]{\mathrm{product}\mathrm{of}n\mathrm{observations}}$

$$\begin{gathered} \Rightarrow \left(\frac{a_1+a_2+a_3+...+a_n}{n}\right)\ge {\left(a_1·a_2·a_3·...·a_n\right)}^{\frac{1}{n}} \\ \Rightarrow \frac{a_1+a_2+a_3+...+a_n}{n}\ge {\left(1\right)}^{\frac{1}{n}} \\ \Rightarrow \frac{a_1+a_2+a_3+...+a_n}{n}\ge 1 \\ \Rightarrow a_1+a_2+a_3+...+a_n\ge n \end{gathered}$$

Hence, the correct answer is option C.