In which of the following situations, does the list of numbers involved make an arithmetic progression, and why ?
The amount of air present in a cylinder when a vacuum pump removes `1/4` of the air remaining in the cylinder at a time.

Answer

To determine whether the situation described makes an arithmetic progression (AP), we need to analyze how the amount of air in the cylinder changes over time as the vacuum pump removes air.

### Step-by-Step Solution:

1. **Initial Amount of Air**: Let the initial amount of air in the cylinder be \( x \).

2. **First Removal**: The vacuum pump removes \( \frac{1}{4} \) of the air remaining in the cylinder. After the first removal, the amount of air left in the cylinder is:
\[ \text{Remaining air} = x - \frac{1}{4}x = \frac{3}{4}x \]

3. **Second Removal**: In the next operation, the vacuum pump again removes \( \frac{1}{4} \) of the air remaining. Now the remaining air is \( \frac{3}{4}x \), so the amount removed is:
\[ \text{Removed air} = \frac{1}{4} \left(\frac{3}{4}x\right) = \frac{3}{16}x \]
The new amount of air in the cylinder is:
\[ \text{Remaining air} = \frac{3}{4}x - \frac{3}{16}x = \frac{12}{16}x - \frac{3}{16}x = \frac{9}{16}x \]

4. **Third Removal**: The vacuum pump removes \( \frac{1}{4} \) of the air remaining again. The remaining air is now \( \frac{9}{16}x \), so the amount removed is:
\[ \text{Removed air} = \frac{1}{4} \left(\frac{9}{16}x\right) = \frac{9}{64}x \]
The new amount of air in the cylinder is:
\[ \text{Remaining air} = \frac{9}{16}x - \frac{9}{64}x = \frac{36}{64}x - \frac{9}{64}x = \frac{27}{64}x \]

5. **Continuing the Process**: This process continues, with each time the amount of air remaining being multiplied by \( \frac{3}{4} \). The sequence of amounts of air remaining can be expressed as:
\[ x, \frac{3}{4}x, \frac{9}{16}x, \frac{27}{64}x, \ldots \]
This is a geometric progression, not an arithmetic progression.

### Conclusion:
The list of numbers involved in this situation does not make an arithmetic progression because the difference between consecutive terms is not constant. Instead, the amounts of air form a geometric progression where each term is multiplied by \( \frac{3}{4} \).