## Finding the Rules or nth Terms for Quadratic Sequences

A quadratic sequence is generated by any rule of the form The problem is often to find the rule for a particular given sequence.

An example of a quadratic sequence is: 2, 4, 8, 14, 22

When we find the difference line we obtain

 2 4 8 14 22
 2 4 6 8

The difference line is not constant so it cannot be an arithmetic sequence. However we can construct a second difference line – the difference between the differences:

 2 4 8 14 22
 2 4 6 8
 2 2 2

Now we have a constant list.If the first difference line is not constant but the second difference line is, the sequence is a quadratic sequence We start by find the first coefficient, of This is equal to the second difference line divided by 2: We now know the sequences We Form an line (1 4 9 16 25) and find the difference between the original sequence and the terms of this line. This will give us another sequence: an arithmetic sequence.

 2 4 8 14 22
 1 4 9 16 22

The difference is

 1 0 -1 -2 -3

The common difference is -1:

 1 0 -1 -2 -3
 -1 -1 --1 -1

Because the common difference is -1 we know this sequence is a sequence.We construct a -1 times table and compare it with the arithmetic sequence (1):

 -1 -2 --3 -4 -5
 1 0 --1 -2 -3

To get the sequence (1) from the -1 times table we have to add 2 so the arithmetic sequence is We add this to the to get the n th term or rule for the quadratic sequence: Example: Find the rule for the sequence: 5, 9, 17, 29, 45

Construct a first and second difference lines:

 5 9 17 29 45
 4 8 12 16
 4 4 4

The second difference line is 4 so we know it is a 4 divided by sequence.Form a line and find the difference between this and the original quadratic sequence.

 5 9 17 29 45
 2 8 18 32 50

The difference is

 3 1 -1 -3 -5

This is an arithmetic or simple sequence. The common difference for this line is -2 so we havea sequence. Form a -2 times table and find the difference between it and the arithmetic sequence.

 2 -4 -6 -8 -10
 3 1 -1 -3 -5

The difference is 5 so the arithmetic sequence is Add this to the to get the formula for the n th term: 