Question

If the arithmetic and geometric means of $a$ and $b$ be $A$and $G$ respectively, then the value of $A-G$ will be:

• A. $\frac{(a-b)}{2}$
• B. $\frac{(a+b)}{2}$
• C. ${\left[\frac{(\sqrt{a}-\sqrt{b})}{\sqrt{2}}\right]}^2$
• D. $\frac{2ab}{(a+b)}$

Answer 1

The correct option is C

${\left[\frac{(\sqrt{a}-\sqrt{b})}{\sqrt{2}}\right]}^2$

Explanation for the correct option.

Let $a$ and $b$ be the numbers,
Then AM, $A=\frac{(a+b)}{2}$
$G.M,\mathrm{G}=\sqrt{\mathrm{ab}}$
$\begin{array}{rcl}A-G& =& \frac{(a+b)}{2}-\sqrt{ab}\\ & =& \frac{a+b-2\sqrt{ab}}{2}\\ & =& \frac{(\sqrt{a}-\sqrt{b}{)}^2}{2}\\ & =& {\left[\frac{(\sqrt{a}-\sqrt{b})}{\sqrt{2}}\right]}^2\end{array}$
Hence, option C is correct.