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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

$Un=a⁢n2+b⁢n+c$

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 second difference needs to be shown to be constant.

From the second example of a 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 $c$:

The general form of a quadratic sequence is given by

$Un=a⁢n2+b⁢n+c$

The first term is given by

$U1=a+b+c$

The difference between each term is

$Un+1-Un =a⁢(n+1)2+b⁢(n+1)+c -(a⁢n2+b⁢n+c)$ $=a⁢(n2+2⁢n+1) +bn+b+c-a⁢n2-b⁢n-c$ $=a⁢n2+2⁢a⁢n+a+b-a⁢n2$ $=2⁢a⁢n+a+b$ So the difference between terms is a quadratic sequence is $Un+1-Un =2⁢a⁢n+a+b$ & the first difference between terms is $U2-U1=3⁢a+b$ 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 $2a$.

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

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