In which of the following situations , does the list of numbers of involved make an arithmetic progression , and why ?
The amount of air present in a cylinder when a vacumm 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 will analyze the amount of air remaining in the cylinder after each removal of air by the vacuum pump.

### Step-by-Step Solution:

1. **Initial Amount of Air**:
Let's assume the initial amount of air in the cylinder is 100 units.

2. **First Removal**:
The vacuum pump removes 1/4 of the air remaining.
- Amount removed = \( \frac{1}{4} \times 100 = 25 \) units.
- Amount of air left after the first removal = \( 100 - 25 = 75 \) units.

3. **Second Removal**:
Now, the vacuum pump removes 1/4 of the remaining air (which is now 75 units).
- Amount removed = \( \frac{1}{4} \times 75 = 18.75 \) units.
- Amount of air left after the second removal = \( 75 - 18.75 = 56.25 \) units.

4. **Third Removal**:
Again, the vacuum pump removes 1/4 of the remaining air (which is now 56.25 units).
- Amount removed = \( \frac{1}{4} \times 56.25 = 14.0625 \) units.
- Amount of air left after the third removal = \( 56.25 - 14.0625 = 42.1875 \) units.

5. **List of Amounts Remaining**:
After three removals, the amounts of air remaining are:
- First removal: 75 units
- Second removal: 56.25 units
- Third removal: 42.1875 units

6. **Checking for Arithmetic Progression**:
In an arithmetic progression, the difference between consecutive terms must be constant.
- Difference between the first and second terms: \( 75 - 56.25 = 18.75 \)
- Difference between the second and third terms: \( 56.25 - 42.1875 = 14.0625 \)

7. **Conclusion**:
Since the differences (18.75 and 14.0625) are not equal, the sequence of amounts of air remaining does not form an arithmetic progression.

### Final Answer:
The situation does not make an arithmetic progression because the differences between the amounts of air remaining after each removal are not constant.

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