Quadratic Sequences - a MaS Tutor.

Prerequisite Materials

A quadratic sequence has $n^{2}$ and no higher powers of $n$ in its $n^{th}$ term expression ( $U_{n}$ ). The general form of a quadratic sequence is given by

U_{n} = a n^{2} + b n + c

where $a$ , $b$ and $c$ 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

$n$	1		2		3		4		5
The $n^{th}$ term, $U_{n}$	3		6		13		24		39
The first difference, $U_{n + 1} - U_{n}$		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 $a$ is given by $a = \frac{second difference}{2}$ .

There are two methods for finding the other two characteristic values $b$ and $c$ :

use properties of the quadratic sequence alone with how to find $a$ given above to determine $b$ and $c$ ;
memorise a set of simultaneous equations to find $a$ , $b$ and $c$ ;
use a method for determining the values of $a$ , $b$ and $c$ by building on the work already completed.