SequencesAQA GCSE Maths: Revision notes

Section 1

What are term-to-term and position-to-term rules?

A sequence is an ordered list of numbers following a pattern. There are two main ways to describe sequences:

Term-to-term rule (also called a recurrence relation):

Describes how to find the next term from the previous term
Example: 'add 5 each time' or 'multiply by 2'
Useful for generating sequences when you know the starting value
Written as: $a_{n+1} = a_n + d$ (for arithmetic sequences)

Position-to-term rule (also called the nth term rule):

Gives a formula to find any term directly using its position
Example: $a_n = 3n + 2$ means the nth term equals 3n + 2
More efficient for finding terms far along the sequence
Allows you to jump straight to any term without calculating all previous ones

Generating terms using rules:

Identify whether you have a term-to-term or position-to-term rule
If term-to-term: start with the first term and apply the rule repeatedly
If position-to-term: substitute the position number into the formula

Key termssequenceterm-to-term ruleposition-to-term rulenth term rule

Example

For the sequence 3, 5, 7, 9, ... the term-to-term rule is 'add 2' and the position-to-term rule is $a_n = 2n + 1$ . Using the formula: when n = 1, $a_1 = 2(1) + 1 = 3$ ; when n = 5, $a_5 = 2(5) + 1 = 11$ .

Section 2

What are special number sequences?

Certain sequences have particular mathematical significance and appear frequently in exam questions:

Triangular numbers:

Sequence: 1, 3, 6, 10, 15, 21, ...
Pattern: 1, 1+2, 1+2+3, 1+2+3+4, ...
nth term rule: $T_n = \frac{n(n+1)}{2}$
Represent dots arranged in triangular patterns

Square numbers:

Sequence: 1, 4, 9, 16, 25, 36, ...
Pattern: $1^2, 2^2, 3^2, 4^2, 5^2, 6^2$ , ...
nth term rule: $S_n = n^2$

Cube numbers:

Sequence: 1, 8, 27, 64, 125, 216, ...
Pattern: $1^3, 2^3, 3^3, 4^3, 5^3, 6^3$ , ...
nth term rule: $C_n = n^3$

Fibonacci-type sequences:

Each term is the sum of the two previous terms
Most famous: 1, 1, 2, 3, 5, 8, 13, 21, ...
Term-to-term rule: $a_{n+1} = a_n + a_{n-1}$
Requires two starting values to generate the sequence

Key termstriangular numberssquare numberscube numbersFibonacci-type sequences

Exam tip

Examiners expect you to recognise these sequences instantly. Learn their formulas and first few terms—you may be asked to identify which type a sequence is without being told.

Common mistake

Students often confuse triangular numbers with square numbers. Remember: triangular numbers grow more slowly (1, 3, 6, 10...) while square numbers are perfect squares (1, 4, 9, 16...).

Section 3

What defines arithmetic sequences and how do you find the nth term?

An arithmetic sequence (or arithmetic progression) is a sequence where the difference between consecutive terms is always the same.

Key features:

Common difference (d): the constant amount added between each term
Can be positive (increasing sequence) or negative (decreasing sequence)
Examples: 2, 5, 8, 11, ... (d = 3) or 20, 17, 14, 11, ... (d = −3)

Finding the nth term rule:

For an arithmetic sequence with first term $a$ and common difference $d$ : $a_n = a + (n-1)d$

Where:

$a_n$ is the nth term
$a$ is the first term
$d$ is the common difference
$n$ is the position

Step-by-step process:

Identify the first term ( $a$ )
Find the common difference ( $d$ ) by subtracting any term from the next term
Substitute into the formula $a_n = a + (n-1)d$
Simplify to get the nth term rule
Use this rule to find any term in the sequence

Key termsarithmetic sequencearithmetic progressioncommon difference

Example

For the sequence 7, 11, 15, 19, ...: first term a = 7, common difference d = 4. The nth term rule is $a_n = 7 + (n-1)4 = 7 + 4n - 4 = 4n + 3$ . To find the 10th term: $a_{10} = 4(10) + 3 = 43$ .

Exam tip

Always simplify your nth term rule fully. Examiners expect a clean final answer like $a_n = 3n + 2$ , not $a_n = 5 + (n-1)3$ .

Section 4

What are geometric sequences and how do you find the nth term?

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant factor, called the common ratio.

Key features:

Common ratio (r): the constant multiplier between consecutive terms
Each term = previous term × r
Can increase rapidly (if r > 1) or decrease towards zero (if 0 < r < 1)
Examples: 2, 6, 18, 54, ... (r = 3) or 100, 50, 25, 12.5, ... (r = 0.5)

Finding the nth term rule:

For a geometric sequence with first term $a$ and common ratio $r$ : $a_n = ar^{n-1}$

Where:

$a_n$ is the nth term
$a$ is the first term
$r$ is the common ratio
$n$ is the position

Step-by-step process:

Identify the first term ( $a$ )
Find the common ratio ( $r$ ) by dividing any term by the previous term
Substitute into the formula $a_n = ar^{n-1}$
Use this rule to find any term

Note: This is a Higher Tier topic—foundation tier students should understand what geometric sequences are but may not be required to derive or use the nth term formula.

Key termsgeometric sequencecommon ratio

Example

For the sequence 3, 12, 48, 192, ...: first term a = 3, common ratio r = 4. The nth term rule is $a_n = 3 × 4^{n-1}$ . To find the 5th term: $a_5 = 3 × 4^4 = 3 × 256 = 768$ .

Think of it like this

In arithmetic sequences, you add the same amount each time (like climbing stairs with equal steps). In geometric sequences, you multiply by the same factor each time (like compound interest or exponential growth).

Section 5

How do you distinguish between arithmetic and geometric sequences?

Examiners frequently ask students to identify whether a sequence is arithmetic, geometric, or neither. Use these methods to tell them apart:

Feature	Arithmetic Sequence	Geometric Sequence
Pattern	Add/subtract the same amount each time	Multiply/divide by the same factor each time
Common value	Common difference (d)	Common ratio (r)
How to find it	Subtract any term from the next: d = $a_{n+1} - a_n$	Divide any term by the previous: r = $\frac{a_{n+1}}{a_n}$
Growth rate	Linear (steady increase/decrease)	Exponential (rapid increase or decrease)
Example	5, 8, 11, 14, ... (d = 3)	2, 6, 18, 54, ... (r = 3)
nth term formula	$a_n = a + (n-1)d$	$a_n = ar^{n-1}$

Testing method:

Calculate differences between consecutive terms: if all differences are equal, it's arithmetic
Calculate ratios between consecutive terms: if all ratios are equal, it's geometric
If neither differences nor ratios are constant, it's a special sequence (e.g. triangular, Fibonacci-type)

Important note: A sequence cannot be both arithmetic and geometric unless it's a constant sequence (e.g. 5, 5, 5, 5, ...) where d = 0 and r = 1.

Key termsdistinguisharithmetic sequencegeometric sequence

Example

Is 3, 6, 12, 24, ... arithmetic or geometric? Check differences: 6−3=3, 12−6=6, 24−12=12 (not constant). Check ratios: 6÷3=2, 12÷6=2, 24÷12=2 (all equal). Therefore it's geometric with r = 2.

Common mistake

Students sometimes mix up the test: they find the differences when they should find ratios (or vice versa). Always try both methods if you're unsure; one will produce a constant value.

Must Know

Generate sequences using term-to-term rules by applying the rule repeatedly from a starting value, and using position-to-term rules by substituting the position number into the formula
Special sequences to recognise: triangular numbers ( $T_n = \frac{n(n+1)}{2}$ ), square numbers ( $n^2$ ), cube numbers ( $n^3$ ), and Fibonacci-type sequences (where each term is the sum of the previous two)
Arithmetic sequences have a constant common difference (d); find the nth term using $a_n = a + (n-1)d$ , where a is the first term
Geometric sequences have a constant common ratio (r); find the nth term using $a_n = ar^{n-1}$ (Higher Tier)
Distinguish sequences by checking if differences between consecutive terms are constant (arithmetic) or if ratios are constant (geometric)
Always simplify nth term rules to their cleanest form, and remember that the first term is at position n = 1, not n = 0

Key termssequencearithmeticgeometricnth termcommon differencecommon ratio

That's the notes covered.

Carry on to the next subtopic.