Physics and Mathematics
What is A.P.
1. Introduction
In many real-life situations, numbers increase or decrease by a fixed amount.
For example:
- Saving ₹500 every month
- Seats in a stadium increasing row-wise
- Marks increasing uniformly
Such number patterns are called an Arithmetic Progression.
2. Definition of Arithmetic Progression
A sequence of numbers is called an Arithmetic Progression (AP) if the difference between any two consecutive terms is always the same constant.
3. General Form of an AP
An arithmetic progression is written as:
[a, a + d, a + 2d, a + 3d, …]
Where:
- a = first term
- d = common difference
4. Common Difference
The common difference is obtained by subtracting any term from the term that follows it.
[d = T₂ − T₁ = T₃ − T₂ = …]
5. Examples with Solutions
Example 1.
[2, 5, 8, 11, 14, …]
Here:
First term [a = 2]
Common difference [d = 3]
Example 2.
[10, 7, 4, 1, −2, …]
Here:
First term [a = 10]
Common difference [d = −3]
Example 3.
[5, 5, 5, 5, …]
Here:
First term [a = 5]
Common difference [d = 0]
This is also a valid AP.
6. Important Observations
- The common difference can be positive, negative, or zero.
- An AP can be increasing, decreasing, or constant.
- Every AP is a sequence, but every sequence is not an AP.
7. How to Check Whether a Sequence is an AP?
Step 1. Find the difference between consecutive terms.
Step 2. If the difference is the same throughout, it is an AP.
Step 3. If the difference changes, it is not an AP.
8. Conceptual Questions with Solutions
1. What is the defining feature of an arithmetic progression?
The defining feature of an AP is the constant common difference between consecutive terms.
2. Can the common difference be negative?
Yes, if the common difference is negative, the AP becomes a decreasing AP.
3. Is [5, 5, 5, 5, …] an AP?
Yes. The common difference is 0, so it is a valid AP.
4. Is every sequence an arithmetic progression?
No. Only sequences with a constant difference are arithmetic progressions.
5. Can an AP have fractional terms?
Yes. An AP can have fractions or decimals as its terms.
6. What happens if the common difference is zero?
All terms become equal, forming a constant AP.
7. Is [1, 4, 9, 16, …] an AP?
No. The differences are not constant, so it is not an AP.
8. Can an AP have negative terms?
Yes. Terms of an AP can be negative.
9. What is meant by increasing AP?
An AP with a positive common difference is called an increasing AP.
10. What is meant by decreasing AP?
An AP with a negative common difference is called a decreasing AP.
11. Is [0, 0, 0, …] an AP?
Yes. It is a constant AP with common difference [0].
12. Why is common difference important?
It helps in identifying and constructing an AP.
13. Can an AP be infinite?
Yes. Most APs are infinite sequences.
14. Can an AP start with zero?
Yes. Zero can be the first term of an AP.
15. Why is AP important in mathematics?
APs are used in number theory, finance, and real-life modeling.
9. FAQs / Common Misconceptions
1. The common difference must be positive.
False. It can be positive, negative, or zero.
2. An AP cannot have repeated terms.
Wrong. If [d = 0], all terms repeat.
3. Fractions cannot form an AP.
Incorrect. Fractions can form a valid AP.
4. Every pattern of numbers is an AP.
False. Only patterns with a constant difference are APs.
5. APs are only increasing.
No. They can also be decreasing or constant.
6. Negative numbers cannot be part of an AP.
Wrong. Negative numbers are allowed.
7. An AP must start from 1.
False. It can start from any real number.
8. Common difference changes with terms.
Incorrect. It must remain constant.
9. Zero common difference means no AP.
Wrong. It gives a constant AP.
10. APs are not useful.
False. They are very important in mathematics and real life.
10. Practice Questions with Step-by-Step Solutions
Question 1. Check whether the sequence [2, 5, 8, 11, 14] is an AP. If yes, find the common difference.
Step-by-Step Solution:
Differences: [5 − 2 = 3], [8 − 5 = 3], [11 − 8 = 3]
All differences are equal.
Conclusion:
The sequence is an AP with common difference [d = 3].
Question 2. Determine whether [10, 7, 4, 1, −2] is an AP.
Step-by-Step Solution:
Differences: [7 − 10 = −3], [4 − 7 = −3], [1 − 4 = −3]
Differences are constant.
Conclusion:
The sequence is an AP with [d = −3].
Question 3. Check whether [1, 4, 9, 16, 25] is an AP.
Step-by-Step Solution:
Differences: [4 − 1 = 3], [9 − 4 = 5], [16 − 9 = 7]
Differences are not equal.
Conclusion:
The sequence is not an AP.
Question 4. Is the sequence [5, 5, 5, 5, 5] an AP?
Step-by-Step Solution:
Differences: [5 − 5 = 0] for all consecutive terms
Difference is constant.
Conclusion:
The sequence is an AP with common difference [d = 0].
Question 5. Check whether [3/2, 5/2, 7/2, 9/2] is an AP.
Step-by-Step Solution:
Differences: [5/2 − 3/2 = 1], [7/2 − 5/2 = 1], [9/2 − 7/2 = 1]
Differences are equal.
Conclusion:
The sequence is an AP with [d = 1].
Question 6. Determine whether [0, −2, −4, −6, −8] is an AP.
Step-by-Step Solution:
Differences: [−2 − 0 = −2], [−4 − (−2) = −2], [−6 − (−4) = −2]
Difference is constant.
Conclusion:
The sequence is an AP with [d = −2].
Question 7. Check whether [2, 4, 8, 16] is an AP.
Step-by-Step Solution:
Differences: [4 − 2 = 2], [8 − 4 = 4], [16 − 8 = 8]
Differences are not constant.
Conclusion:
The sequence is not an AP.
Question 8. Is the sequence [−1, −3, −5, −7] an AP?
Step-by-Step Solution:
Differences: [−3 − (−1) = −2], [−5 − (−3) = −2], [−7 − (−5) = −2]
Differences are equal.
Conclusion:
The sequence is an AP with [d = −2].
Question 9. Check whether [1, 1/2, 1/3, 1/4] is an AP.
Step-by-Step Solution:
Differences: [1/2 − 1 = −1/2], [1/3 − 1/2 ≠ −1/2]
Differences are not the same.
Conclusion:
The sequence is not an AP.
Question 10. Determine whether [7, 10, 13, 16] is an AP.
Step-by-Step Solution:
Differences: [10 − 7 = 3], [13 − 10 = 3], [16 − 13 = 3]
All differences are equal.
Conclusion:
The sequence is an AP with common difference [d = 3].
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