### Prerequisite Materials

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

$${U}_{n}=a{n}^{2}+bn+c$$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 ${n}^{\mathrm{th}}$ 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 $a=\frac{\mathrm{second\; difference}}{2}$.

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.