Question

Let x be the A.M. and y,z be the two G.M. between any two positive number then prove that $\frac{y^3+z^3}{xyz}=2$.

Answer 1

Given,

$a,y,z,b$ are in G.P.

$\Rightarrow yz=ab$

$y^2=az\Rightarrow y^3=ayz$

$z^2=yb\Rightarrow z^3=ayz$

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

Given,

$\frac{y^3+z^3}{xyz}$

$=\frac{2[ayz+byz]}{(a+b)yz}$

$=\frac{2\left[\right(a+b\left)yz\right]}{(a+b)yz}$

$=2$

Hence proved.