Quadratic Sequences

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Prerequisite Materials

Arithmetic Sequence

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

Un =an2+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 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 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:

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