📚 Question
In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
(ii) 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.
📌 Given Information
- A vacuum pump removes 1⁄4 of the air remaining in the cylinder each time
- This means 3⁄4 of the air remains after each operation
- We need to check if the amounts of air form an AP
🎯 To Find
Whether the amount of air remaining in the cylinder forms an Arithmetic Progression (AP) and justify the answer.
💡 Key Concept
Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant.
\[a_2 – a_1 = a_3 – a_2 = a_4 – a_3 = d\]
Geometric Progression (GP): A sequence where the ratio between consecutive terms is constant.
\[\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = r\]
✍️ Step-by-Step Solution
Step 1: Assume initial amount of air
Let the initial amount of air in the cylinder be \(V\) (say, 1 unit for simplicity).
Initial amount: \(V = 1\) unit
Step 2: Calculate air remaining after each pump operation
Since the pump removes 1⁄4 of the air each time, 3⁄4 of the air remains:
Initially: \(a_1 = 1\)
After 1st operation: \(a_2 = 1 \times \frac{3}{4} = \frac{3}{4}\)
After 2nd operation: \(a_3 = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16}\)
After 3rd operation: \(a_4 = \frac{9}{16} \times \frac{3}{4} = \frac{27}{64}\)
After 4th operation: \(a_5 = \frac{27}{64} \times \frac{3}{4} = \frac{81}{256}\)
Sequence: \[1, \frac{3}{4}, \frac{9}{16}, \frac{27}{64}, \frac{81}{256}, \ldots\]
Step 3: Check if differences are constant (AP test)
Calculate the differences between consecutive terms:
\(a_2 – a_1 = \frac{3}{4} – 1 = -\frac{1}{4}\)
\(a_3 – a_2 = \frac{9}{16} – \frac{3}{4} = \frac{9}{16} – \frac{12}{16} = -\frac{3}{16}\)
\(a_4 – a_3 = \frac{27}{64} – \frac{9}{16} = \frac{27}{64} – \frac{36}{64} = -\frac{9}{64}\)
Observation: The differences are NOT constant: \(-\frac{1}{4} \neq -\frac{3}{16} \neq -\frac{9}{64}\)
Step 4: Check if ratios are constant (GP test)
Calculate the ratios between consecutive terms:
\(\frac{a_2}{a_1} = \frac{3/4}{1} = \frac{3}{4}\)
\(\frac{a_3}{a_2} = \frac{9/16}{3/4} = \frac{9}{16} \times \frac{4}{3} = \frac{3}{4}\)
\(\frac{a_4}{a_3} = \frac{27/64}{9/16} = \frac{27}{64} \times \frac{16}{9} = \frac{3}{4}\)
Observation: The ratios ARE constant: \(\frac{3}{4}\). This is a Geometric Progression (GP), not an AP!
Step 5: Conclusion
Since the difference between consecutive terms is NOT constant, the amount of air does NOT form an Arithmetic Progression.
However, it forms a Geometric Progression with common ratio \(r = \frac{3}{4}\).
✅ Final Answer
No, the amount of air does NOT form an Arithmetic Progression.
The differences between consecutive terms are not constant. Instead, it forms a Geometric Progression with common ratio r = 3/4.
🔍 Why It’s NOT an AP
For a sequence to be an AP, the difference between consecutive terms must be constant.
In this case:
- First difference: \(-\frac{1}{4}\)
- Second difference: \(-\frac{3}{16}\)
- Third difference: \(-\frac{9}{64}\)
Since \(-\frac{1}{4} \neq -\frac{3}{16} \neq -\frac{9}{64}\), the differences are NOT constant, so it’s NOT an AP.
🔄 Understanding Geometric Progression
This situation represents a multiplicative decrease, which creates a Geometric Progression:
General term of this GP:
\[a_n = 1 \times \left(\frac{3}{4}\right)^{n-1}\]
Where \(n\) is the number of operations (starting from \(n = 1\) for the initial amount).
⚠️ Common Mistakes to Avoid
- Confusing AP with GP: Remember, AP has constant difference, while GP has constant ratio.
- Not calculating all differences: Always check multiple consecutive differences to confirm the pattern.
- Misunderstanding “removes 1/4”: If 1/4 is removed, then 3/4 remains—don’t confuse these values.
- Assuming all sequences are AP: Not every sequence is an AP; some are GP, and some follow other patterns.
📝 Practice Problems
- A tank has 1000 liters of water. If 1⁄5 of the water is removed each time, does the remaining water form an AP? If not, what type of progression is it?
- The value of a car depreciates by 10% each year. Does the value form an AP or GP over the years?
- Compare: (a) Removing 100 liters each time vs. (b) Removing 10% each time. Which forms an AP and which forms a GP?
- If a pump removes a fixed amount of air (say, 50 ml) each time instead of a fraction, would it form an AP?
🔗 Related Topics
Written by Farhan Mansuri
M.Sc. Mathematics | B.Ed. | 15+ Years Teaching Experience
Farhan Mansuri is a dedicated mathematics educator with over 15 years of experience teaching CBSE curriculum. He specializes in making complex mathematical concepts accessible to Class 10 students.
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