x

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

From the second example of a 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 $c$:

The general form of a quadratic sequence is given by

$${U}_{n}=a{n}^{2}+bn+c$$The first term is given by

$${U}_{1}=a+b+c$$The difference between each term is

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

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

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