Algebra

Sequences: Linear, Quadratic, Geometric, and Fibonacci

Year 10 · Year 11

✓Identify and distinguish between linear, quadratic, geometric, and Fibonacci sequences.
✓Determine the nth term of a linear sequence.
✓Determine the nth term of a quadratic sequence (Higher Tier only).
✓Calculate any term in a linear, quadratic, geometric, or Fibonacci sequence.
✓Use the nth term to check if a given number is a term in a sequence.

Key concepts

Linear Sequences

A linear sequence (also known as an arithmetic sequence) is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference. For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3.

The nth term is of the form an + b, where 'a' is the common difference and 'b' is a constant.

Quadratic Sequences

A quadratic sequence is a sequence where the first differences between consecutive terms form a linear sequence, and the second differences between consecutive terms are constant. These sequences are typically covered at Higher Tier GCSE.

The nth term is of the form an^2 + bn + c, where 'a', 'b', and 'c' are constants. The constant second difference is equal to 2a.

Geometric Sequences

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 6, 18, 54, ..., the common ratio is 3.

Fibonacci Sequences

A Fibonacci sequence is a special type of sequence where each term is the sum of the two preceding terms. The most common Fibonacci sequence starts with 0, 1, 1, 2, 3, 5, ... or 1, 1, 2, 3, 5, ...

F_n = F_{n-1} + F_{n-2} for n > 2, with initial terms F_1 and F_2.

Key facts to remember

1A linear sequence has a constant first difference. Its nth term is of the form an + b.
2A quadratic sequence has a constant second difference. Its nth term is of the form an^2 + bn + c.
3In a quadratic sequence, the constant second difference is equal to 2a.
4A geometric sequence has a constant common ratio (each term is found by multiplying the previous term by this ratio).
5A Fibonacci sequence is defined by adding the two preceding terms to get the next term.
6To find the nth term of a linear sequence, identify the common difference ('a') and then adjust for the constant ('b').

Worked examples

Example 1

Find the nth term of the linear sequence 5, 8, 11, 14, ...

IFind the common difference between consecutive terms:

II8 - 5 = 3

III11 - 8 = 3

IV14 - 11 = 3

VThe common difference is 3.

VIThis means the nth term will start with 3n.

VIICompare 3n with the original sequence:

VIIIFor n=1, 3n = 3. Original term is 5. (5 - 3 = 2)

9For n=2, 3n = 6. Original term is 8. (8 - 6 = 2)

10For n=3, 3n = 9. Original term is 11. (11 - 9 = 2)

11The difference is always +2.

12Therefore, the nth term is 3n + 2.

Answer

3n + 2

To check, substitute n=4: 3(4) + 2 = 12 + 2 = 14, which is the 4th term.

Example 2

Find the nth term of the quadratic sequence 2, 7, 16, 29, ... (Higher Tier)

IFind the first differences:

II7 - 2 = 5

III16 - 7 = 9

IV29 - 16 = 13

VFirst differences: 5, 9, 13

VIFind the second differences:

VII9 - 5 = 4

VIII13 - 9 = 4

9Second differences: 4, 4

10Since the second difference is constant, it's a quadratic sequence. The second difference is equal to 2a.

112a = 4

12a = 2

13So, the nth term starts with 2n^2. Let's write out the terms for 2n^2:

14For n=1: 2(1)^2 = 2

15For n=2: 2(2)^2 = 8

16For n=3: 2(3)^2 = 18

17For n=4: 2(4)^2 = 32

18Sequence 2n^2: 2, 8, 18, 32, ...

19Subtract the 2n^2 terms from the original sequence terms:

20Original sequence: 2, 7, 16, 29

212n^2 sequence: 2, 8, 18, 32

22Difference: (2-2), (7-8), (16-18), (29-32)

23Difference sequence: 0, -1, -2, -3, ...

24Find the nth term of this new linear sequence (0, -1, -2, -3, ...):

25Common difference is -1. So it's -n.

26For n=1, -n = -1. Original term is 0. (0 - (-1) = 1)

27So, the nth term for this linear part is -n + 1.

28Combine the 2n^2 part and the linear part:

29The nth term is 2n^2 - n + 1.

Answer

2n^2 - n + 1

Always check your formula by substituting a few values of n. For n=3: 2(3)^2 - 3 + 1 = 2(9) - 3 + 1 = 18 - 3 + 1 = 16, which is correct.

Example 3

A geometric sequence begins 3, 6, 12, ... Find the 5th term of the sequence.

IFind the common ratio by dividing a term by its preceding term:

II6 / 3 = 2

III12 / 6 = 2

IVThe common ratio is 2.

VList the given terms and continue the pattern:

VI1st term: 3

VII2nd term: 6

VIII3rd term: 12

94th term: 12 * 2 = 24

105th term: 24 * 2 = 48

Answer

For GCSE, you are expected to find terms by repeatedly multiplying by the common ratio, rather than using the ar^(n-1) formula directly.

Example 4

A sequence is defined by u_1 = 2, u_2 = 3, and u_{n+2} = u_{n+1} + u_n for n >= 1. Find the first 5 terms of this sequence.

IWrite down the given initial terms:

IIu_1 = 2

IIIu_2 = 3

IVUse the recursive rule u_{n+2} = u_{n+1} + u_n to find subsequent terms:

VFor n=1: u_3 = u_2 + u_1 = 3 + 2 = 5

VIFor n=2: u_4 = u_3 + u_2 = 5 + 3 = 8

VIIFor n=3: u_5 = u_4 + u_3 = 8 + 5 = 13

Answer

2, 3, 5, 8, 13

Fibonacci sequences are defined by their recursive relationship, not an nth term formula.

Common mistakes

✗Confusing the common difference of a linear sequence with the common ratio of a geometric sequence.
✗Making arithmetic errors when calculating differences or terms, especially with negative numbers.
✗For quadratic sequences, forgetting that the second difference is 2a, not a.
✗Not checking the derived nth term formula by substituting n=1, 2, 3 to ensure it generates the correct initial terms.
✗Incorrectly identifying the type of sequence, leading to using the wrong method.

Exam tips

★Always show your working for finding differences (first and second) as this can earn method marks, even if your final nth term is incorrect.
★After finding an nth term formula, substitute n=1, 2, 3 back into your formula to verify it generates the original sequence terms.
★For geometric and Fibonacci sequences, clearly state the common ratio or the recursive rule you are using to find subsequent terms.
★Read the question carefully to determine if you need to find the nth term, a specific term (e.g., the 10th term), or determine if a number is part of the sequence.

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