Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (B)

$2$

Answer 2

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, $\frac{y^{3}+z^{3}}{xyz}$ $=\frac{a^{3}r^{3}+a^{3}r^{6}}{\frac{\left(a+b\right)}{2} \cdot a^{2} r^{3}}$ $= \frac{2a\left(1+r^{3}\right)}{a+ar^{3}} = 2$.