Question

If $a_1,a_2,...a_n$ are positive numbers such that $a_1\times a_2\times ...\times a_n=1$, then their sum is

• A. a positive integer
• B. divisible by $\mathrm{n}$
• C. never less than $\mathrm{n}$
• D. none of these

Answer 1

The correct option is C

never less than $\mathrm{n}$

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

We know that Arithmetical mean $\ge$Geometrical mean

$\begin{array}{rcl}AM& \ge & GM\\ \frac{(a_1+a_2+.....\dots a_n)}{n}& \ge & {(a_1.a_2.a_3\dots a_n)}^{1/n}\\ & & \\ \frac{(a_1+a_2+.....\dots a_n)}{n}& \ge & (1{)}^{1/n}\\ \frac{(a_1+a_2+.....\dots a_n)}{n}& \ge & 1\\ (a_1+a_2+.....\dots a_n)& \ge & n\end{array}$( given ${\mathit{a}}_{\mathbf{1}}\mathbf{\times }{\mathit{a}}_{\mathbf{2}}\mathbf{\times }\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\times }{\mathit{a}}_{\mathbf{n}}\mathbf{=}\mathbf{1}$)

So, the sum is never less than $\mathrm{n}$.

Hence, the correct option is (C).