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. $\left(\frac{\sqrt a-\sqrt b}{\sqrt2}\right)^2$

Answer 1

Answer: (D)

$P= \frac{a+b}{2} , Q= \sqrt{ab}$
$ \therefore 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}$