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 $y, z$ be the two geometric means between any two positive numbers.
Then value of $\frac{y^{3} + z^{3}}{x y z}$ is

A
2
B
3
C
$\frac{1}{2}$
D
$\frac{3}{2}$

Solution:

Let two positive numbers be $a$ and $b .$ Then $x = (a + b) / 2 .$ Also, $a, y, z, b$ are in G.P.

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

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

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