Question

If $x$ be the arithmetic mean and $y,z$ be two geometric means between two positive numbers, then the value of $\frac{y^3+z^3}{xyz}$ is

Answer 1

Assuming the two numbers be $a$ and $b.$
Now, we know that,
$x=\frac{a+b}{2}\cdots \left(1\right)$
And $a,y,z,b$ are in G.P. so,
$r={\left(\frac{b}{a}\right)}^{1/3}y=a{\left(\frac{b}{a}\right)}^{1/3}\cdots \left(2\right)z=a{\left(\frac{b}{a}\right)}^{2/3}\cdots \left(3\right)$

Now, using equation $\left(1\right),\left(2\right)$ and $\left(3\right),$
$\frac{y^3+z^3}{xyz}=\frac{a^3\times \frac{b}{a}+a^3\times \frac{b^2}{a^2}}{ab\left(\frac{a+b}{2}\right)}=\frac{2(a^{2}b+a{b}^2)}{a^{2}b+a{b}^2}=2$