Class 10 Maths · Chapter 5 · Topic 1 of 4
An AP adds the same amount each time
Test equal consecutive differences before using AP formulas.
Ideas you’ll use
- Arithmetic progression (AP)
- A list in which each term after the first is found by adding the same fixed number d. This number is the common difference.
Understand it, one idea at a time
Choose an idea. Follow its reason, calculation and clue in the picture.
Equal jumps versus doubling
Term value (number)
Term number n (integer)
- AP: add 3 · Original sequence, evaluated at the displayed integer positions.
- Not an AP: double · Original sequence, evaluated at the displayed integer positions.
Assumptions: Only the dots at positive integer term numbers belong to the sequence. Positions between dots are not extra terms.
Read the plotted values
AP: add 3
| Term number n (integer) | Term value (number) |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |
| 4 | 13 |
| 5 | 16 |
Not an AP: double
| Term number n (integer) | Term value (number) |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
The AP has equal vertical differences. Doubling creates different increases.
Compare neighbours
Subtract each term from the next.
4,7,10,13: differences +3In the picture: Equal jumps appear between every consecutive dot.
Read all the reasoning together
Compare neighbours
Subtract each term from the next.
4,7,10,13: differences +3Allow negative and zero jumps
An AP can decrease or stay constant.
15,11,7,3: d=−4Reject changing differences
A doubling list has a pattern but is not an AP.
2,4,8,16: differences 2,4,8
Test equal consecutive differences before using AP formulas.
Worked example: A pattern is not automatically an AP
2, 4, 8, 16 has differences 2, 4, 8The differences change, so this doubling pattern is not an AP. Once a list is known to be an AP, any consecutive pair gives d. To test an unfamiliar list, compare the available consecutive differences.
Read the supporting explanation
For 4, 7, 10, 13, … the differences are all +3. For 15, 11, 7, 3, … they are all −4. A constant list is an AP with d = 0. A finite AP has a last term; an infinite AP does not.
Go deeper: What does the list tell us about the future?
A few numbers can fit many possible rules. Here, writing “AP” or stating a fixed increase tells us how the pattern continues. Do not assume every real-world quantity continues forever.
Your turn
Try the idea yourself. Show your reasoning, then compare.
Question
Is 1.5, 1.0, 0.5, 0 an AP? Give d and the next term.
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