Question

Let $a$ be the arithmetic mean and $b, c$ be the twio geometric means between two positive numbers. Find the value of $\frac{b^{3}+c^{3}}{a b c}$.

Answer 1

Let the two numbers be $x$ and $y$.
$\Rightarrow a=\frac{x+y}{2} $
$x, b, c, y $ are in G.P.
$y=x r^{3} $
$\Rightarrow r=\left(\frac{y}{x}\right)^{\frac{1}{3}}$
$\Rightarrow \frac{b^{3}+c^{3}}{a b c}=\frac{x^{3} r^{3}+x^{3} r^{6}}{\left(\frac{x+y}{2}\right)(x r)\left(x r^{2}\right)}=2$