The same step every time
Arithmetic progression
A growing pattern
A pattern of tiny squares grows stage by stage. The number of squares is 1, 5, 9, 13, … At each stage, 4 squares get added to the corners of the earlier pattern.
Look at 1, 5, 9, 13. What is special about how the numbers grow?
What this lesson covers
The idea
An arithmetic progression (AP) is a sequence in which the difference between consecutive terms is constant; this constant is the common difference d, which can also be negative.
A growing pattern
A pattern of tiny squares grows stage by stage. The number of squares is 1, 5, 9, 13, … At each stage, 4 squares get added to the corners of the earlier pattern.
Look at 1, 5, 9, 13. What is special about how the numbers grow?
- The same number is added every time
- A different number is added every time
- Each number is double the one before
Is the gap constant?
For each list, press Show the differences. Then tell whether it is an AP: is the gap always the same?
Constant difference
An arithmetic progression (AP) is a sequence in which the difference between consecutive terms is constant. This constant is the common difference d, and it can also be negative.
1, 5, 9, 13, 17, … is an AP with d = 4. And 11, 7, 3, −1, −5, … is an AP with d = −4: the terms decrease by 4 each time.
Notes
An arithmetic progression (AP) is a sequence in which the difference between consecutive terms is constant: the common difference d, which can also be negative.
Check yourself
Which of these is an AP?
What is the common difference d of the AP 20, 17, 14, 11, …?
Answer: -3
d = 17 − 20 = −3. The terms decrease by 3 each time, so d is negative.
An AP starts 4, 9, 14, … What is the next term?
Answer: 19
d = 9 − 4 = 5, so the next term is 14 + 5 = 19.
Is 1, 3, 6, 10, 15, … an AP?
- 2, 4, 8, 16, …. The differences are 2, 4, 8, …, which are not the same. (Each term doubles; that is a different kind of sequence.)
- 1, 4, 9, 16, …. The differences are 3, 5, 7, …, which keep changing. Not an AP.
- 5, 8, 11, 14, … — correct. Yes. The difference is 3 every time, so it is an AP with d = 3.
- No, the differences 2, 3, 4, 5 are not all equal — correct. Right. The differences 2, 3, 4, 5 change, so it is not an AP. These are the triangular numbers.
- Yes, the numbers keep growing. Growing is not enough. An AP needs the same difference every time.
- Yes, the differences 2, 3, 4, 5 go up steadily. The difference must be constant, not just regular. Here it changes at every step.