Let x be the arithmetic mean and y, z be the two geometric means between any two positive numbers.Then value of y3+z3xyz is

# Let $x$ be the arithmetic mean and  be the two geometric means between any two positive numbers.Then value of $\frac{{y}^{3}+{z}^{3}}{xyz}$ is

1. A

2

2. B

3

3. C

$\frac{1}{2}$

4. D

$\frac{3}{2}$

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### Solution:

Let two positive numbers be $a$ and $b.$ Then Also,  are in G.P.

If $r$ is the common ratio of this G.P…, then $b=a{r}^{3}⇒r=\left(b/a{\right)}^{1/3}.$

We have $\frac{{y}^{3}+{z}^{3}}{xyz}=\frac{{a}^{3}{r}^{3}+{a}^{3}{r}^{6}}{x\left(ar\right)\left(a{r}^{2}\right)}=\frac{a\left(1+{r}^{3}\right)}{x}=\frac{a+b}{\left(a+b\right)/2}=2$

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