Question

Let \( x \) be the arithmetic mean and \( y, z \) be the two geometric means between any two positive numbers, then \[ \frac{y^3 + z^3}{x y z} = \text{----------} \]

• A. \( \frac{1}{3} \)
• B. \( 1 \)
• C. \( \frac{1}{7} \)
• D. \( 2 \)

Hint:

When solving problems with arithmetic and geometric means, make sure to use their standard definitions and simplify expressions step by step.

Answer 1

The Correct Option is D

We are given that \( x \) is the arithmetic mean and \( y \) and \( z \) are the geometric means between two positive numbers. Let's express these means as follows: Step 1: Define the means - The arithmetic mean \( x \) between two numbers \( a \) and \( b \) is given by: \[ x = \frac{a + b}{2} \] - The geometric means \( y \) and \( z \) between the same two numbers are the numbers that satisfy: \[ y = \sqrt{ab} \quad \text{and} \quad z = \sqrt{ab} \] Step 2: Use the properties of means We are asked to find the value of: \[ \frac{y^3 + z^3}{xyz} \] Using the values of \( y \) and \( z \), we can simplify this expression. Step 3: Simplify the expression We know that \( y = z = \sqrt{ab} \), so we have: \[ y^3 + z^3 = 2y^3 = 2 (\sqrt{ab})^3 = 2a^{3/2} b^{3/2} \] And for the denominator: \[ xyz = (\sqrt{ab})^3 = a^{3/2} b^{3/2} \] Thus: \[ \frac{y^3 + z^3}{xyz} = \frac{2a^{3/2} b^{3/2}}{a^{3/2} b^{3/2}} = 2 \] Thus, the correct answer is \( \boxed{2} \).