## 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, ofThis is equal to the second difference line divided by 2:

We now know the sequencesWeForm anline(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 25

The difference is

1 0 -1 -2 -3 (1)

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 asequence.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 thetoget the n ^{th } term or rule for the quadratic sequence:

Example: Find the rule forthe 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 oversequence.Form aline 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 haveasequence. 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 isAdd this to theto get the formula for the n ^{th } term: