Sequences: Linear and Quadratic

A sequence is an ordered list of numbers that follows a rule. Understanding sequences and being able to find their rules is a key part of AQA GCSE Mathematics. Foundation students need to work with linear sequences, while Higher tier students must also handle quadratic and geometric sequences.

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Key Vocabulary

Term	Meaning	Example
Term	Each number in a sequence	In 2, 5, 8, 11... each number is a term
Position (n)	The place of a term in the sequence (1st, 2nd, 3rd...)	n = 1 for the first term
Term-to-term rule	A rule that describes how to get from one term to the next	Add 3
Position-to-term rule (nth term)	A formula giving the value of any term from its position	3n - 1
Common difference	The constant difference between consecutive terms in a linear sequence	In 2, 5, 8, 11: d = 3

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Term-to-Term Rules

The simplest way to describe a sequence is by saying what you do to get from one term to the next.

Worked Example 1

Find the next three terms: 4, 7, 10, 13, ...

The term-to-term rule is add 3.

Next three terms: 16, 19, 22

Worked Example 2

Find the next three terms: 1, 2, 4, 8, ...

The term-to-term rule is multiply by 2.

Next three terms: 16, 32, 64

Exam Tip: The term-to-term rule is useful for finding the next few terms, but it is not efficient for finding, say, the 100th term. For that, you need the nth term formula.

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The nth Term of a Linear Sequence

A linear sequence has a constant difference between consecutive terms. Its nth term formula has the form:

nth term = dn + (a - d)

where d is the common difference and a is the first term.

Steps to Find the nth Term

1. Find the common difference (d).
2. Write dn (this gives the "times table" part).
3. Compare the sequence to the d times table and find the adjustment needed.

Worked Example 3

Find the nth term of: 5, 8, 11, 14, ...

Step 1: Common difference = 3

Step 2: The 3 times table is 3, 6, 9, 12, ...

Step 3: Compare:

Position (n)	1	2	3	4
Sequence	5	8	11	14
3n	3	6	9	12
Difference	+2	+2	+2	+2

The sequence is always 2 more than 3n.

Answer: nth term = 3n + 2

Check: When n = 1: 3(1) + 2 = 5. When n = 4: 3(4) + 2 = 14. Correct.

Worked Example 4

Find the nth term of: 7, 3, -1, -5, ...

Common difference = -4

4n = 4, 8, 12, 16 but we need -4n = -4, -8, -12, -16

Compare:

Position (n)	1	2	3	4
Sequence	7	3	-1	-5
-4n	-4	-8	-12	-16
Difference	+11	+11	+11	+11

Answer: nth term = -4n + 11

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Using the nth Term

Once you have the nth term, you can:

• Find any term: substitute the position number.
• Determine if a number is in the sequence: set the nth term equal to that number and solve for n. If n is a positive whole number, the number is in the sequence.

Worked Example 5

The nth term of a sequence is 5n - 3.

(a) Find the 50th term.

5(50) - 3 = 250 - 3 = 247

(b) Is 138 in the sequence?

5n - 3 = 138

5n = 141

n = 28.2

Since n is not a whole number, 138 is not in the sequence.

(c) Is 147 in the sequence?

5n - 3 = 147

5n = 150

n = 30

Since n = 30 is a positive whole number, 147 is the 30th term.

Exam Tip: When asked "Is the number N in this sequence?", set the nth term formula equal to N and solve for n. You must clearly state whether n is a whole number and what that means.

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The nth Term of a Quadratic Sequence [H]

A quadratic sequence has a constant second difference (the differences between consecutive differences are constant). Its nth term has the form:

an^2 + bn + c

Steps to Find the nth Term

1. Find the first differences.
2. Find the second differences (these should be constant).
3. The coefficient of n^2 is half the second difference: a = (second difference) / 2.
4. Subtract an^2 from each term to get a linear sequence.
5. Find the nth term of that linear sequence (dn + e).
6. Combine: nth term = an^2 + dn + e.

Worked Example 6

Find the nth term of: 3, 9, 19, 33, 51, ...

Step 1: First differences: 6, 10, 14, 18

Step 2: Second differences: 4, 4, 4 (constant, so it IS quadratic)

Step 3: a = 4/2 = 2, so the n^2 term is 2n^2

Step 4: Subtract 2n^2 from each term:

n	1	2	3	4	5
Sequence	3	9	19	33	51
2n^2	2	8	18	32	50
Difference	1	1	1	1	1

Step 5: The remaining sequence is 1, 1, 1, 1, 1 — this is a constant: +1.

Answer: nth term = 2n^2 + 1

Worked Example 7

Find the nth term of: 2, 9, 20, 35, 54, ...

First differences: 7, 11, 15, 19

Second differences: 4, 4, 4

a = 4/2 = 2

Subtract 2n^2:

n	1	2	3	4	5
Sequence	2	9	20	35	54
2n^2	2	8	18	32	50
Remainder	0	1	2	3	4

The remaining sequence is 0, 1, 2, 3, 4 which has nth term = n - 1.

Answer: nth term = 2n^2 + n - 1

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Geometric Sequences [H]

A geometric sequence has a constant common ratio between consecutive terms (each term is multiplied by the same number to get the next term).

Worked Example 8

Find the common ratio and the next two terms: 3, 6, 12, 24, ...

Common ratio = 6/3 = 2

Next two terms: 24 x 2 = 48, 48 x 2 = 96

Worked Example 9