A quadratic sequence has and no higher powers of in its term expression (). 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 difference between terms needs to be shown to be an arithmetic sequence, i.e., have a common difference.
From the second example of a quadratic 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 :
- use properties of the quadratic sequence alone with how to find given above to determine and ;
- memorise a set of simultaneous equations to find , and ;
- use a method for determining the values of , and by building on the work already completed.