Question

If the A.M., G.M. and H.M. between two positive numbers $a$ and $b$ are equal, then

• A. $a=b$
• B. $ab=1$
• C. $a>b$
• D. $a<b$

Answer 1

The correct option is A $a=b$

A] $AM=\frac{a+b}{2}GM=\sqrt{ab}HM=\frac{2ab}{a+b}$

AM = GM AM = HM

$\frac{(a+b)}{2}=\sqrt{ab}$ $\frac{a+b}{2}=\frac{2ab}{a+b}$

= $(a+b{)}^2=4ab=(a+b{)}^2=4ab$

HM = GM

$\frac{2ab}{a+b}=\sqrt{ab}$

= $4ab=(a+b{)}^2$

= $a^2+b^2+2ab=4ab$

= $a^2+b^2-2ab=0$

= $(a-b{)}^2=0$

$\therefore a=b$