Sequences

You will learn to:

- Find the nth term of linear and quadratic sequences
- find any term in a sequence using the nth term rule.

Method

There are many types of sequence, but the ones that you are most likely to come across are: - Linear - Quadratic

Example of a linear sequence:

        2, 4, 6, 8, 10, 12

Example of a quadratic sequence:

        1, 4, 9, 16, 25, 36

You may be asked to find the nth term of a sequence. This is a term-to-term rule that applies to the whole sequence. Lets use the first sequence as an example.

n	1	2	3	4	5	6
Sequence	2	4	6	8	10	12

The nth term is how you get from n to the sequence. In this case, the nth term rule is 2n, as you are doubling n each time.

Worked Example

    Find the nth term, and 25th term of the sequence:
            -1, 1, 3, 5, 7, 9
    1. Find the diffeence between each term.
        1 - -1 = 2
        3 -  1 = 2
        5 -  3 = 2
           ...
    This number is the coefficient of n, as they are all the same.

2n	2	4	6	8	10	12
Sequence	-1	1	3	5	7	9

How do we go from 2 to -1? Take away 3. How do we go from 4 to 1? Minus 3.

Nth term = 2n - 3.

25th term = 2 * 25 - 3 = 47.

Quadratics

In a quadratic sequence, the first differences aren’t the same, and the nth term takes the form an² + bn + c

Worked Example.

    Find the nth term, and the 13th term of the following sequence.
            -1, 5, 15, 29 47.

    Find the first differences.
           -1 +6 = 5 , + 10 = 15, + 14 = 29, + 18 =  47

    Find the second differences.
            +6 +4 = +10, +4 = +14, +4 = +18

    The second diffference is double the coefficient of n<sup>2</sup>, ie 2n<sup>2</sup>.

2n2	2	8	18	32	50
Sequence	-1	5	15	28	47
2n2 --> Sequence	-3	-3	-3	-3	-3

The nth term = 2n² The 13th term = 2 x 13² - 3 = 335