If the arithmetic geometric and harmonic menas between two positive real numbers be A, G and H, then

Options

A. `A^(2) = GH`
B. `H^(2) = AG`
C. `G = AH`
D. `G^(2) = AH`

Answer

To solve the problem, we need to establish the relationships between the Arithmetic Mean (A), Geometric Mean (G), and Harmonic Mean (H) of two positive real numbers \( a \) and \( b \).

### Step-by-step Solution:

1. **Define the Means**:
- The Arithmetic Mean (A) of \( a \) and \( b \) is given by:
\[ A = \frac{a + b}{2} \]
- The Geometric Mean (G) of \( a \) and \( b \) is given by:
\[ G = \sqrt{ab} \]
- The Harmonic Mean (H) of \( a \) and \( b \) is given by:
\[ H = \frac{2ab}{a + b} \]

2. **Square the Geometric Mean**:
- We start by squaring the Geometric Mean:
\[ G^2 = ab \]
- This will be our Equation 1.

3. **Relate the Arithmetic Mean and Harmonic Mean**:
- We can express the relationship between the Arithmetic Mean and Harmonic Mean using the following:
\[ A \cdot H = A \cdot \frac{2ab}{a + b} \]
- This will be our Equation 2.

4. **Equate the Two Expressions**:
- From Equation 1, we have \( G^2 = ab \).
- From Equation 2, we can express \( A \cdot H \) as:
\[ A \cdot H = \left(\frac{a + b}{2}\right) \cdot \left(\frac{2ab}{a + b}\right) = ab \]
- Since both expressions equal \( ab \), we can set them equal to each other:
\[ G^2 = A \cdot H \]

5. **Conclusion**:
- Therefore, we have established that:
\[ G^2 = A \cdot H \]
- This means that the relationship between the Arithmetic Mean, Geometric Mean, and Harmonic Mean is:
\[ G^2 = A \cdot H \]

### Final Result:
The correct relationship is \( G^2 = A \cdot H \).

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