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Q.
Let $x$ be the arithmetic mean and $y, z$ be the two geometric means between any two positive numbers. The value of $\frac{y^{3}+ z^{3}}{xyz}$ is
Sequences and Series
Solution:
Let the two numbers be $a$ and $b$.
$ x = A.M. = \frac{a+b}{2} $
$ a, y = ar, z= ar^{2}, b= ar^{3}$ are in $G. P$.
Now, $\frac{y^{3}+z^{3}}{xyz} $
$=\frac{a^{3}r^{3}+a^{3}r^{6}}{\frac{\left(a+b\right)}{2} \cdot a^{2} r^{3}}$
$ = \frac{2a\left(1+r^{3}\right)}{a+ar^{3}} = 2 $.