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Maths Gcse Aqa Higher

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Exam code:8300

Quadratic sequences

What is a quadratic sequence?

  • A quadratic sequence has an n th term formula that involves n2

  • The second differences are constant (the same)

    • These are the differences between the first differences

    • For example, 3, 9, 19, 33, 51, …
      1st Differences: 6, 10, 14, 18, …

      2nd Differences: 4, 4, 4, …

  • The sequence with the n th term formula n2 are the square numbers 

    • 1, 4, 9, 16, 25, 36, 49, …

      • From 12, 22, 32, 42, …

How do I find the nth term formula for any quadratic sequence?

  • STEP 1
    Work out the sequences of first and second differences

    • e.g. for the sequence 1, 10, 23, 40, 61

sequence

1

10

23

40

61

first difference

+9

+13

+17

+21

second difference

+4

+4

+4

  • STEP 2
    Divide the second difference by 2 to find the coefficient of n2

    • e.g. a = 4 ÷ 2 = 2

  • STEP 3
    Write out the first three or four terms of an2 and subtract the terms from the corresponding terms of the given sequence

    • e.g. for the sequence 1, 10, 23, 40, 61

sequence

1

10

23

40

2n2

2

8

18

32

difference

-1

2

5

8

  • STEP 4
    Work out the nth term of these differences to find the bn + c part of the formula

    • e.g. the nth term of -1, 2, 5, 8, … is bn + c = 3n − 4

  • STEP 5
    Find an2 + bn + c by adding together this linear nth term to an2 term

    • e.g. an2 + bn + c = 2n2 + 3n − 4

Examiner Tips and Tricks

  • You must learn the square numbers from 12 to 152

Worked Example

For the sequence 5, 7, 11, 17, 25, ….

(a) Find a formula for the nth term.
Start by finding the first and second differences

Sequence: 5, 7, 11, 17, 25

First differences: 2, 4, 6, 8, …

Second difference: 2, 2, 2, …

Hence 

a = 2 ÷ 2 = 1

Now write down an2 (just n2 in this case as a = 1) and subtract the terms from the original sequence

sequence: 5, 7, 11, 17, …

an2. : 1, 4, 9, 16, …

 difference: 4, 3, 2, 1, …

Work out the nth term of these differences to give you bn + c

bn + c = −n + 5

Add an2 and bn + c together to give you the nth term of the sequence

nth term = n2 − n + 5

(b) Hence find the 20th term of the sequence.

Substitute n = 20 into n2 − n + 5

(20)2 − 20 + 5 = 400 − 15 

20th term = 385

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