Question

If the arithmetic, geometric and harmonic means between two distinct positive real numbers be $A,G$ and $H$ respectively, then the relation between them is:

• A. $A>G>H$
• B. $A>G<H$
• C. $H>G>A$
• D. $G>A>H$

Answer 1

The correct option is A

$A>G>H$

Explanation for the correct option.

Step 1: Relation between $A\&G$

Given, $A,G$ and $H$ are arithmetic, geometric and harmonic means between two distinct positive real numbers.

Then, using the definition of arithmetic, geometric and harmonic means we get:

$\mathrm{A}=\frac{(\mathrm{a}+\mathrm{b})}{2}$,$G=\sqrt{ab}$ and $H=\frac{2ab}{(a+b)}$.
$\begin{array}{rcl}A-G& =& \frac{(a+b)}{2}-\sqrt{ab}\\ & =& \frac{(a+b-2\sqrt{ab})}{2}\\ & =& \frac{{\left(\sqrt{a}-\sqrt{b}\right)}^2}{2}\ge 0\end{array}$

So $A\ge G$

Step 2: Relation between $G\&H$
$\begin{array}{rcl}G-H& =& \sqrt{ab}-\frac{2ab}{(a+b)}\\ & =& \frac{\sqrt{ab}[a+b-2\sqrt{\left(ab\right)]}}{(a+b)}\\ & =& \frac{\sqrt{(}ab\left)\right(\sqrt{a}-\sqrt{b}{)}^2}{(a+b)}\ge 0\\ G& \ge & H\\ & & \\ & & \end{array}$
Thus $\mathrm{A}\ge \mathrm{G}\ge \mathrm{H}$.
Hence option A is correct.