There are a number of different types of sequences, we have already looked at linear sequences where the number goes up or down by the same each time.
We need to now be able to work with two more types of sequence.
Geometric: A seqeunce that has a common ratio. This means we multiply by the same amount each time.
Quadratic: A quadratic sequnce has a common second difference. This means that the difference between the differences will be the same.
Example 1: Describe the rule for, and give the next two terms of the sequence 3, 6, 12, 24...
We should notice that the sequence is doubling each time, this means the rule (the common ratio) is x 2
If we then continue the sequence, the next two terms are 48 and 96.
Example 2: Find the next term in the sequence 3, 6, 11, 18, 27
We first have to identify the sequence type here. It isn't going up by the same (so it can't be linear), it isn't multiplying each time, so it cant be a geometric sequence. This leaves a quadratic.
Step 1: Find the difference between the terms.
The difference between the first ans second terms is 3
The difference between the second and third terms is 5
The difference between the third and fourth terms is 7
The difference between the fourth and fifth terms is 9
Do you see how the difference is going up by 2 each time? this means...
The difference between the fifth and sixth terms is 11
This means the next term in our sequence 3, 6, 11, 18, 27 will be 38