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

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:

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 sequencesWe Form 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.

The difference is

The common difference is -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):

To get the sequence (1) from the -1 times table we have to add 2 so the arithmetic sequence is We add this to theto 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:

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

The difference is

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.

The difference is 5 so the arithmetic sequence isAdd this to theto get the formula for the n ^{th term:}