If a, b, c, d are in harmonical progrssion such that `agtd,` then which one of the following is correct ?

Options

A. `a+c=b+d`
B. `a+cgtb+d`
C. `ac=bd`
D. `ab=cd`

Explanation

To solve the problem, we need to analyze the relationship between the terms \(a\), \(b\), \(c\), and \(d\) when they are in harmonic progression (HP) and given that \(a > d\).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression (HP)**:
- If \(a\), \(b\), \(c\), and \(d\) are in HP, then the reciprocals \( \frac{1}{a} \), \( \frac{1}{b} \), \( \frac{1}{c} \), and \( \frac{1}{d} \) are in arithmetic progression (AP).

2. **Using the Property of HP**:
- For four terms \(a\), \(b\), \(c\), and \(d\) in HP, the relationship can be expressed as:
\[ \frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b} = \frac{1}{d} - \frac{1}{c} \]
- This implies that:
\[ 2b = \frac{a+c}{2} \quad \text{and} \quad 2c = \frac{b+d}{2} \]

3. **Given Condition**:
- We know that \(a > d\). This implies that the first term \(a\) is greater than the last term \(d\).

4. **Finding Relationships**:
- From the properties of HP, we can derive:
\[ b = \frac{2ac}{a+c} \quad \text{and} \quad c = \frac{2bd}{b+d} \]
- We can express \(A + C\) and \(B + D\) using these relationships.

5. **Comparing \(A + C\) and \(B + D\)**:
- We need to find the ratio:
\[ \frac{A + C}{B + D} \]
- Substituting the values, we get:
\[ A + C = a + c \quad \text{and} \quad B + D = b + d \]

6. **Establishing the Inequality**:
- Since \(a > d\) and the terms are in HP, we can conclude that:
\[ A + C > B + D \]
- Therefore, we can state:
\[ A + C > B + D \]

7. **Conclusion**:
- The correct relationship is:
\[ A + C > B + D \]

### Final Answer:
The correct option is: \(A + C > B + D\).

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