If a be the arithmetic mean and b, c be the two geometric means between any two positive numbers, then `(b^3 + c^3)` / abc equals

Options

A. `(ab)^(1//2) //c`
B. 1
C. `a^2c//b`
D. 2

Answer

To solve the problem, we need to find the value of \((b^3 + c^3) / (abc)\) given that \(a\) is the arithmetic mean and \(b\) and \(c\) are the two geometric means between two positive numbers \(x\) and \(y\).

### Step-by-Step Solution:

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

2. **Define the Geometric Means (GMs)**:
Since \(b\) and \(c\) are the geometric means between \(x\) and \(y\), we can express them in terms of \(x\) and \(y\). The geometric means can be defined as:
\[ b = x \cdot r \quad \text{and} \quad c = x \cdot r^2 \]
where \(r\) is the common ratio.

3. **Determine the Common Ratio**:
Since \(b\) and \(c\) are the geometric means, we can express \(r\) in terms of \(x\) and \(y\):
\[ r = \left(\frac{y}{x}\right)^{1/3} \]
Therefore,
\[ b = x \left(\frac{y}{x}\right)^{1/3} = x^{2/3} y^{1/3} \]
\[ c = x \left(\frac{y}{x}\right)^{2/3} = x^{1/3} y^{2/3} \]

4. **Calculate \(b^3 + c^3\)**:
\[ b^3 = (x^{2/3} y^{1/3})^3 = x^2 y \]
\[ c^3 = (x^{1/3} y^{2/3})^3 = x y^2 \]
Therefore,
\[ b^3 + c^3 = x^2 y + x y^2 \]

5. **Calculate \(abc\)**:
\[ abc = a \cdot b \cdot c = \left(\frac{x + y}{2}\right) \cdot (x^{2/3} y^{1/3}) \cdot (x^{1/3} y^{2/3}) \]
Simplifying \(b \cdot c\):
\[ b \cdot c = x^{2/3} y^{1/3} \cdot x^{1/3} y^{2/3} = x^{1} y^{1} = xy \]
Thus,
\[ abc = \frac{x + y}{2} \cdot xy \]

6. **Substituting into the Expression**:
Now we substitute \(b^3 + c^3\) and \(abc\) into the expression:
\[ \frac{b^3 + c^3}{abc} = \frac{x^2 y + x y^2}{\frac{x + y}{2} \cdot xy} \]
This simplifies to:
\[ = \frac{x^2 y + x y^2}{\frac{xy (x + y)}{2}} = \frac{2(x^2 y + x y^2)}{xy (x + y)} \]

7. **Factor the Numerator**:
The numerator can be factored:
\[ = \frac{2xy(x + y)}{xy(x + y)} = 2 \]

### Final Answer:
Thus, the value of \(\frac{b^3 + c^3}{abc}\) is:
\[ \boxed{2} \]