A quadratic sequence has and no higher powers of in its term (). The general form of a quadratic sequence is given by
where , and are numbers that can be thought of as characterising the quadratic sequence and offen are to be found.
An example of a quadratic sequence is 1, 4, 9, 16, 25, ... which happens to be the square numbers. Another is 3, 6, 13, 24, 39, ... which is less easily identified as a particular sequence. To determine if a given sequence is indeed a quadratic sequence the second difference needs to be shown to be constant.
From the second example of a sequence
|The first difference,||3||7||11||15|
|The second difference||4||4||4|
The second difference is a constant of 4, therefore showing that the sequence given is an example of a quadratic sequence. It can be shown that is given by .
There are two methods for finding the other two characteristic values and :
The general form of a quadratic sequence is given by
The first term is given by
The difference between each term is
So the difference between terms is a quadratic sequence is & the first difference between terms is Now taking the difference of the differences between terms, or, sometimes called, the second difference, that is, a difference minus the previous difference,
So the difference between the differences in terms (or the second difference) is the constant .
We can use these results to help determine the values of a, b & c.