IF p q r s are in harmonic progression and `p gt s` then

Options

A. `1/(ps) lt 1/(qr)`
B. `q+r=p+s`
C. `1/q+1/9=1/r+1/s`
D. none

Answer

To solve the problem, we need to analyze the condition given that \( p, q, r, s \) are in harmonic progression (HP) and \( p > s \).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression**:
- If \( p, q, r, s \) are in HP, then their reciprocals \( \frac{1}{p}, \frac{1}{q}, \frac{1}{r}, \frac{1}{s} \) are in arithmetic progression (AP).

2. **Setting Up the AP Condition**:
- For four terms in AP, the condition can be expressed as:
\[ 2 \cdot \text{(middle term)} = \text{(first term)} + \text{(last term)} \]
- In our case, we can express this as:
\[ 1/q - 1/p = 1/s - 1/r \]

3. **Rearranging the Equation**:
- We can rearrange the terms:
\[ \frac{1}{q} + \frac{1}{r} = \frac{1}{s} + \frac{1}{p} \]

4. **Interpreting the Condition**:
- This equation indicates a relationship between the reciprocals of \( p, q, r, \) and \( s \). Since \( p > s \), we can infer that the relationship must hold true under this condition.

5. **Conclusion**:
- The derived condition \( \frac{1}{q} + \frac{1}{r} = \frac{1}{s} + \frac{1}{p} \) is the final result we were looking for.

### Final Result:
Thus, the condition that holds true if \( p, q, r, s \) are in harmonic progression and \( p > s \) is:
\[ \frac{1}{q} + \frac{1}{r} = \frac{1}{s} + \frac{1}{p} \]