Question

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

Answer 1

Answer: 2

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