Quadratic Sequences

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 first 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)
If the second differences there are constant, we know that the example is a quadratic sequence

What should we be able to 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 n^th term of a quadratic sequence in terms of n
This formula will be in the form:
n^th term = an² + bn + c
(The process for finding a, b, and c is given below)

How do I find the n^th term of 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!
- e.g. sequence: 1, 10, 23, 40, 61
  - first difference: 9, 13, 17, 21, ...
  - second differences: 4, 4, 4, ...
a = [the second difference] ÷ 2
- e.g. a = 4 ÷ 2 = 2
Write out the first three or four terms of an² with the first three or four terms of the given sequence underneath.
Work out the difference between each term of an² and the corresponding term of the given sequence.
- e.g. an² = 2n² = 2, 8, 18, 32, ...
  sequence = 1, 10, 23, 40, ...
  difference = -1, 2, 5, 8, ...
Work out the linear nth term of these differences. This is bn + c.
- e.g. bn + c = 3n − 4
Add this linear nth term to an². Now you have an² + bn + c.
- e.g. an² + bn + c = 2n² + 3n − 4

Examiner Tip

Before doing the very formal process to find the n^thterm, try comparing the sequence to the square numbers 1, 4, 9, 16, 25, … and see if you can spot the formula
For example:
- Sequence 4, 7, 12, 19, 28, …
- Square Numbers 1 4 9 16 25
- We can see that each term of the sequence is 3 more than the equivalent square number so the formula is
  n^th term = n²+ 3
- This could save you a lot of time!

Worked example

For the sequence 5, 7, 11, 17, 25, ....

(a)

Find a formula for the n^thterm.

Start by finding the first and second differences

Sequence: 5, 7, 11, 17, 25

First differences: 2, 4, 6, 8, ...

Second difference: 2, 2, 2, ...

Hence

a = 2 ÷ 2 = 1

Now write down an² (just n² in this case as a = 1) with the sequence underneath, and on the next line write the difference between an² and the sequence

an^2.: 1, 4, 9, 16, ...

sequence: 5, 7, 11, 17, ...

difference: 4, 3, 2, 1, ...

Work out the nth term of these differences to give you bn + c

bn + c = −n + 5

Add an² and bn + c together to give you the n^th term of the sequence

n^th term = n² − n + 5

(b)

Hence find the 20^th term of the sequence.

Substitute n = 20 into n² − n + 5

(20)² − 20 + 5 = 400 − 15

20^th term = 385

Quadratic Sequences (AQA GCSE Maths)

Revision Note