Quadratic Sequences

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A quadratic sequence has n2 and no higher powers of n in its nth term (Un). The general form of a quadratic sequence is given by


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 nth term, Un 3 6 13 24 39
The first difference, Un+1-Un 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=second difference2.

There are two methods for finding the other two characteristic values b and c:

The general form of a quadratic sequence is given by


The first term is given by


The difference between each term is

Un+1-Un =a(n+1)2+b(n+1)+c -(an2+bn+c) =a(n2+2n+1) +bn+b+c-an2-bn-c =an2+2an+a+b-an2 =2an+a+b So the difference between terms is a quadratic sequence is Un+1-Un =2an+a+b & the first difference between terms is U2-U1=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.

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