Question

If $P,Q,R$ be the $A.M.,G.M.,H.M.$, respectively between any two rational numbers a and b, then $P−Q$ is

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

Answer 1

Answer: (D)

$P=\frac{a+b}{2},Q=\sqrt{ab}$
$∴P−Q=\frac{a+b}{2}−\sqrt{ab}$
$=\frac{{\left(\sqrt{a}−\sqrt{b}\right)}^2}{2}={\left(\frac{\sqrt{a}−\sqrt{b}}{\sqrt{2}}\right)}^2$