Linear, Quadratic, and Exponential Models
Arithmetic progressions
Find the common difference, the nth term and the sum of an arithmetic progression.
An arithmetic progression is a list of numbers where you get from one to the next by adding the same amount every time. 3, 7, 11, 15 is one: the step is always 4. That step is called the common difference, and it is the only thing you need to know about the list besides where it starts.
Why the formula says (n - 1)
The nth term is a + (n - 1)d. The first term takes no steps at all, the second takes one step, the third takes two — so the tenth term has taken nine. Getting that off-by-one right is most of the skill.
Have a play
This line is the progression 5, 10, 15, 20 and onwards. Move from one term to the next and count the steps you have taken.
Tap anywhere on the line.
Worked example
What is the 10th term of 3, 7, 11, 15, ...?
The first term a is 3, and the common difference d is 7 - 3 = 4.
Always check the difference on more than one pair: 11 - 7 is also 4, so it really is an AP.
Try it together
Now find the sum of the first 10 terms of 2, 4, 6, 8, ...
There is a lovely trick for sums. Pair the first term with the last, the second with the second-to-last, and so on — every pair adds up to the same total.
1.First, what is the common difference of 2, 4, 6, 8, ...?
Have a go
Have a go on your own: what is the 6th term of 4, 9, 14, ...?
Hint: The difference is 5, and the 6th term is five steps along from 4.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.