Quadratic Sequences.

A quadratic sequence has $n^{2}$ and no higher powers of $n$ in its $n^{th}$ term ( $U_{n}$ ). 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 second difference needs to be shown to be constant.

From the second example of a 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 $c$ :

The general form of a quadratic sequence is given by

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

The first term is given by

U_{1} = a + b + c

The difference between each term is

$U_{n + 1} - U_{n} = a {(n + 1)}^{2} + b (n + 1) + c - (a n^{2} + b n + c)$ $= a (n^{2} + 2 n + 1) + b n + b + c - a n^{2} - b n - c$ $= a n^{2} + 2 a n + a + b - a n^{2}$ $= 2 a n + a + b$ So the difference between terms is a quadratic sequence is $U_{n + 1} - U_{n} = 2 a n + a + b$ & the first difference between terms is $U_{2} - U_{1} = 3 a + b$ Now taking the difference of the differences between terms, or, sometimes called, the second difference, that is, a difference minus the previous difference, $(U_{n + 2} - U_{n + 1}) - (U_{n + 1} - U_{n})$ $= (2 a (n + 1) + a + b) - (2 a n + a + b)$ $= 2 a$

So the difference between the differences in terms (or the second difference) is the constant $2 a$ .

We can use these results to help determine the values of a, b & c.