Let x be the arithmetic mean and y,z be the two geometric means between any two positive numbers. Then, value of `(y^3+z^3)/xyz` ​=?

Explanation

To solve the problem, we need to find the value of \((y^3 + z^3) / (xyz)\), where \(x\) is the arithmetic mean and \(y\) and \(z\) are the two geometric means between two positive numbers \(A\) and \(B\).

### Step-by-Step Solution:

1. **Define the Arithmetic Mean \(x\)**:
\[ x = \frac{A + B}{2} \]

2. **Define the Geometric Means \(y\) and \(z\)**:
Since \(y\) and \(z\) are the two geometric means between \(A\) and \(B\), we can express them in terms of \(A\) and \(B\). The sequence of numbers is \(A, y, z, B\).

The geometric means can be expressed as:
\[ y = A \cdot r \quad \text{and} \quad z = A \cdot r^2 \]
where \(r\) is the common ratio.

3. **Find the Common Ratio \(r\)**:
The fourth term \(B\) can be expressed as:
\[ B = A \cdot r^3 \]
Therefore, we can solve for \(r\):
\[ r = \left(\frac{B}{A}\right)^{\frac{1}{3}} \]

4. **Substituting for \(y\) and \(z\)**:
Now substituting \(r\) back into the equations for \(y\) and \(z\):
\[ y = A \cdot \left(\frac{B}{A}\right)^{\frac{1}{3}} = A^{\frac{2}{3}} B^{\frac{1}{3}} \]
\[ z = A \cdot \left(\frac{B}{A}\right)^{\frac{2}{3}} = A^{\frac{1}{3}} B^{\frac{2}{3}} \]

5. **Calculate \(y^3 + z^3\)**:
Using the formula for the sum of cubes:
\[ y^3 + z^3 = (y + z)(y^2 - yz + z^2) \]
First, calculate \(y + z\):
\[ y + z = A^{\frac{2}{3}} B^{\frac{1}{3}} + A^{\frac{1}{3}} B^{\frac{2}{3}} \]
Now calculate \(y^2\) and \(z^2\):
\[ y^2 = (A^{\frac{2}{3}} B^{\frac{1}{3}})^2 = A^{\frac{4}{3}} B^{\frac{2}{3}}, \quad z^2 = (A^{\frac{1}{3}} B^{\frac{2}{3}})^2 = A^{\frac{2}{3}} B^{\frac{4}{3}} \]
\[ yz = A^{\frac{2}{3}} B^{\frac{1}{3}} \cdot A^{\frac{1}{3}} B^{\frac{2}{3}} = A^{1} B^{1} = AB \]
Thus,
\[ y^2 - yz + z^2 = A^{\frac{4}{3}} B^{\frac{2}{3}} - AB + A^{\frac{2}{3}} B^{\frac{4}{3}} \]

6. **Calculate \(xyz\)**:
\[ xyz = x \cdot y \cdot z = \left(\frac{A + B}{2}\right) \cdot (A^{\frac{2}{3}} B^{\frac{1}{3}}) \cdot (A^{\frac{1}{3}} B^{\frac{2}{3}}) \]
\[ = \frac{A + B}{2} \cdot AB \]

7. **Final Calculation**:
Substitute \(y^3 + z^3\) and \(xyz\) into the expression:
\[ \frac{y^3 + z^3}{xyz} \]
After simplification, we find that:
\[ \frac{y^3 + z^3}{xyz} = 2 \]

### Final Answer:
\[ \frac{y^3 + z^3}{xyz} = 2 \]