Question

Let $x$ be the arithmetic mean and $y,z$ be the two geometric means between any two positive numbers, then $\frac{y^3+z^3}{xyz}=$..............

Answer 1

Let $a$ and $b$ are two positive numbers,

Thus $x=\frac{a+b}{2}$ (arithmetic mean)

If $y,z$ are two geometric means between $a,b$

$a,y,z,b$ will be in G.P.

Common ratio

$\left(r\right)={\left(\frac{b}{a}\right)}^{\frac{1}{2+1}}={\left(\frac{b}{a}\right)}^{\frac{1}{3}}$

$y=ar=a\times {\left(\frac{b}{a}\right)}^{\frac{1}{3}}$

$z=a{r}^2=a\times {\left(\frac{b}{a}\right)}^{\frac{2}{3}}$

$\frac{y^3+z^3}{xyz}=\frac{a^3\times \frac{b}{a}+a^3\times {\left(\frac{b}{a}\right)}^2}{\frac{a+b}{2}\times a{\left(\frac{b}{a}\right)}^{\frac{1}{3}}\times a{\left(\frac{b}{a}\right)}^{\frac{2}{3}}}$

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

$=2$