First slide

Means - A.M/G.M/H.M

Question

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

Moderate

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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