Question

If A,G,H be respectively the A.M., G.M. and H.M. between two given quantities a and b, then prove that
A,G,H are in G.P.

Answer 1

$A=\frac{a+b}{2}$

$G=\sqrt{ab}$

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

$A\times H=\frac{a+b}{2}\times \frac{2ab}{a+b}$

$=ab$

$=(\sqrt{ab}{)}^2$

$=G^2$

$\Rightarrow G=\sqrt{A\times H}$

Hence, proved that A,G and H are in G.P.