### Prerequisite Materials

Arithmetic Sequence

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

$Un =a⁢n2+b⁢n+c$

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

 $n$ The second difference The ${n}^{\mathrm{th}}$ term, The first difference, ${U}_{n}$ ${U}_{n+1}-{U}_{n}$ 1 2 3 4 5 3 6 13 24 39 3 7 11 15 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.

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