The amount of air present in cylinder when a vacuum pump remove 1/4 of the air remaining in the cylinder at a time. Q= In this question the question is arithmetic progression is or not and why?

Question

The amount of air present in cylinder when a vacuum pump remove 1/4 of the air remaining in the cylinder at a time.   Q= In this question the question is arithmetic progression  is or not and why?

Answer

As in this question the successive common diff. is not constant.
::It is not an Ap..

Answer

Here u can see that
each time ¼ of the gas is removed
and the air remaining in the container is removed ¼ that the air left
so u can see that there is not a constant common difference
therefore it is not in AP

Answer

Understanding the Problem
In this scenario, a vacuum pump removes 1/4 of the air remaining in a cylinder at a time. This process involves repeated actions that can help us analyze the sequence of air remaining in the cylinder over time.
The Sequence of Air Removal
- Initial Amount: Let's denote the initial amount of air in the cylinder as A.
- First Removal: After the first operation, 1/4 of A is removed, leaving 3/4 of A.
- Subsequent Removals:
- After the second operation, 1/4 of the remaining air (which is now 3/4 A) is removed, leaving (3/4) * (3/4 A) = 9/16 A.
- This process continues, where each time, 1/4 of the remaining air is removed.
Identifying the Pattern
- Mathematical Representation: The remaining air can be represented as:
- 1st removal: (3/4)A
- 2nd removal: (3/4)^2 A
- 3rd removal: (3/4)^3 A
- General Form: After n removals, the remaining air is (3/4)^n A.
Is This an Arithmetic Progression?
- Definition of Arithmetic Progression: An arithmetic progression (AP) is a sequence in which the difference between consecutive terms is constant.
- Analysis: In our case, the difference between terms is not constant. Instead, the ratio of the remaining air after each removal is constant, indicating a geometric progression (GP) instead.
Conclusion
- Final Determination: The process described is not an arithmetic progression; it is a geometric progression due to the constant ratio of the remaining air after each operation. The air decreases in a multiplicative manner rather than an additive one.

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Progressions with equal common difference are known as

Needed a Document for 1 Mark's questions of an arithematic progression? Related: Exercise 1 - Chapter 5 - Arithmetic Progressions, Class 10, Mathematics?

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