What is a sequence?
A Sequence is simply a set of numbers (or objects) in an order.
What is a quadratic sequence?
Unlike in a Linear Sequence, in a Quadratic Sequence the differences between the terms (the first differences) are not constant. However the differences between the differences (the second differences) are constant.
Another way to think about this is that in a Quadratic Sequence, the sequence of differences is a Linear Sequence.
eg Sequence 2, 3, 6, 11, 18, …
1st Differences 1 3 5 7 (a Linear Sequence)
2nd Differences 2 2 2 (Constant)
Because the second differences there are constant, we know that the example is a Quadratic Sequence.
What can we do with quadratic sequences?
You should be able to recognise and continue a Quadratic Sequence.
You should also be able to find a formula for the nth term of a Quadratic Sequence in terms of n. This formula will be in the form
nth term = an2 + bn + c
(The process for finding a, b, and c is given below.)
How to find the nth term formula for a quadratic sequence
- Work out the sequences of first and second differences
- Note: Check that the first differences are not constant and the second differences are constant, to make sure you have a quadratic sequence!
- Use the first and second differences to find a, b, and c in the nth term formula. Follow these steps in order:
- 2a is the 1st second difference (*)
- 3a + b is the 1st first difference
- a + b + c is the 1st term
(*) – or any second difference, as in a Quadratic Sequence they are all the same!)