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By the end of these notes, you should be able to:
A progression is simply a list of numbers arranged in a specific order, where each number follows a rule. Each number in the list is called a term. You may also hear progressions called sequences — they mean the same thing.
For example: 3, 7, 11, 15, 19, ... is a progression. Each term follows a pattern.
There are two main types of progressions you need to know: Arithmetic Progressions and Geometric Progressions.
An arithmetic progression (AP) is a sequence of numbers where you add (or subtract) the same fixed number to get from one term to the next. This fixed number is called the common difference, written as d.
In simple words: If the gap between any two consecutive (next-to-each-other) terms is always the same, it's an AP.
5, 8, 11, 14, 17, ...
The common difference here is d = 3. Because the same number (3) is added every time, this is an arithmetic progression.
20, 15, 10, 5, 0, ...
The common difference here is d = −5. The sequence is going down by 5 each time. It is still an AP — the common difference just happens to be negative.
To find d, subtract any term from the term that comes after it:
d=any term−the term before itExample: In the sequence 3, 7, 11, 15, ...
d=7−3=4(check: 11−7=4✓)If the first term is called a and the common difference is d, then the terms of an AP look like this:
| Position | Term |
|---|---|
| 1st term | a |
| 2nd term | a+d |
| 3rd term | a+2d |
| 4th term | a+3d |
| 5th term | a+4d |
Notice the pattern: the nth term of an AP is:
nth term=a+(n−1)dThis formula is very important. It lets you find any term in the sequence without listing every single one.
Example using the formula:
Find the 10th term of the AP: 5, 8, 11, 14, ...
Here, a=5 and d=3.
10th term=5+(10−1)×3=5+27=32Sign in to view full notes