What is a sequence in mathematics, and how is it related to algebra?

In mathematics, a sequence is an ordered list of numbers that often follows a specific pattern or rule. In algebra, sequences are used to understand patterns and relationships between numbers. Each number in a sequence is called a term, and sequences can be finite or infinite. Understanding sequences is crucial for solving problems related to series and progressions, which are key concepts in the AQA GCSE Mathematics curriculum.

How do you find the nth term of an arithmetic sequence?

To find the nth term of an arithmetic sequence, you use the formula: nth term = a + (n-1)d, where 'a' is the first term and 'd' is the common difference between consecutive terms. This formula helps you calculate any term in the sequence without listing all the terms, which is an important skill in AQA GCSE Mathematics.

What is the difference between arithmetic and geometric sequences?

The main difference between arithmetic and geometric sequences lies in how their terms are generated. In an arithmetic sequence, each term is obtained by adding a constant value, known as the common difference, to the previous term. In contrast, a geometric sequence is formed by multiplying the previous term by a constant value called the common ratio. Understanding these differences is essential for solving sequence-related problems in the AQA GCSE Mathematics syllabus.

How can you determine if a sequence is arithmetic or geometric?

To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. If it is, the sequence is arithmetic. For a geometric sequence, verify if the ratio of consecutive terms is constant. Recognizing these patterns is a fundamental skill in algebra, especially for students preparing for the AQA GCSE Mathematics exams.

What are some real-life applications of sequences in mathematics?

Sequences have numerous real-life applications, such as in financial calculations, computer algorithms, and natural phenomena modeling. For example, arithmetic sequences can model situations involving regular savings or payments, while geometric sequences are used in calculating compound interest or population growth. Understanding these applications helps students appreciate the practical relevance of sequences in the AQA GCSE Mathematics curriculum.

Arithmetic sequence 7, 11, 15, 19, 23 with common difference d = 4

At a glance

A term-to-term rule tells you how to get the next term from the previous one (e.g. 'add 3')
A position-to-term rule (nth term formula) gives any term directly: substitute $n = 1, 2, 3, \ldots$
nth term of a linear (arithmetic) sequence: $T_n = a + (n-1)d = dn + (a - d)$ , where $d$ is the common difference
nth term of a geometric sequence: $T_n = ar^{n-1}$ , where $a$ is the first term and $r$ is the common ratio
nth term of a quadratic sequence has the form $T_n = an^2 + bn + c$ ; find $a$ from half the second difference
Special sequences to recognise: triangular $\tfrac{1}{2}n(n+1)$ , square $n^2$ , cube $n^3$ , Fibonacci (each term = sum of previous two)

What you should be able to do

Generate terms of a sequence from a term-to-term rule or a position-to-term rule (A23)
Recognise and continue arithmetic, geometric, triangular, square, cube and Fibonacci-type sequences (A24)
Find and use the nth term of a linear (arithmetic) sequence (A25)
Find the nth term of a quadratic sequence (A25)
Determine whether a given value is a term of a sequence (A25)
Use the nth term to decide whether a sequence is increasing, decreasing or periodic (A23, A24)

Study notes

Term-to-Term and Position-to-Term Rules

Two ways to define a sequence — AQA spec A23

Every sequence can be described in two equivalent ways.

Term-to-term rule: states how to move from one term to the next.

Example: Start at 3; add 5 each time. Sequence: $3, 8, 13, 18, 23, \ldots$

More complex term-to-term rules involve multiplication or a combination:

Double the previous term: $2, 4, 8, 16, 32, \ldots$
$u_{n+1} = 3u_n - 1$ , $u_1 = 2$ : gives $2, 5, 14, 41, \ldots$

Position-to-term rule (nth term formula): gives the value of any term directly from its position $n$ .

Example: $T_n = 4n - 1$ gives terms $3, 7, 11, 15, \ldots$ (substitute $n = 1, 2, 3, 4, \ldots$ )

Which to use when:

Task	Use
Find the next few terms	Term-to-term
Find the 50th term	Position-to-term (nth term)
Check if 99 is in the sequence	Position-to-term (solve $T_n = 99$ )

Fibonacci-type sequences have the term-to-term rule $u_{n+2} = u_{n+1} + u_n$ . The classic Fibonacci sequence is $1, 1, 2, 3, 5, 8, 13, 21, \ldots$ There is no simple closed-form nth term expected at GCSE; you generate terms by adding the two preceding terms.

A periodic sequence repeats; e.g. $1, 2, 3, 1, 2, 3, \ldots$ has period 3
To find a term-to-term rule, look at the differences between consecutive terms
For a Fibonacci-type sequence you may be given the first two terms and asked to generate further terms

Common pitfall

Students sometimes confuse the term value with the term position. If $T_n = 5n + 2$ then the 3rd term ( $n = 3$ ) is $5(3) + 2 = 17$ — not the term whose value is 3.

Arithmetic (Linear) Sequences

Sequences with a constant difference and their nth term — AQA spec A24, A25

An arithmetic sequence has a constant difference $d$ between consecutive terms.

General form: $a,\; a+d,\; a+2d,\; a+3d,\; \ldots$

nth term formula:

$T_n = a + (n-1)d$

which can be rewritten as $T_n = dn + (a - d)$ .

Example: Find the nth term of $7, 11, 15, 19, \ldots$

Common difference $d = 4$ ; first term $a = 7$ :

$T_n = 4n + (7 - 4) = 4n + 3$

Check: $T_1 = 4(1) + 3 = 7$ ✓; $T_4 = 4(4) + 3 = 19$ ✓

Finding a specific term: $T_{20} = 4(20) + 3 = 83$ .

Is a value in the sequence? Set $T_n = \text{target}$ and solve for $n$ . If $n$ is a positive integer, yes; otherwise no.

Example: Is 100 in the sequence $4n + 3$ ?

$4n + 3 = 100 \implies 4n = 97 \implies n = 24.25$ . Not an integer, so 100 is not in the sequence.

Increasing/decreasing: if $d > 0$ the sequence increases; if $d < 0$ it decreases.

Arithmetic sequence 7, 11, 15, 19, 23 — each pair of consecutive terms is separated by the same common difference d = 4.

The coefficient of $n$ in the nth term formula always equals the common difference $d$
Find $d$ by subtracting any term from the next: $d = T_{k+1} - T_k$
The nth term of $3, 7, 11, \ldots$ is $4n - 1$ (not $4n + 3$ ) — always double-check by substituting $n = 1$

Common pitfall

The most common error is finding the 0th term instead of the 1st. The constant in $dn + c$ is $c = a - d$ , found by extending the sequence one step backwards, not by substituting $n = 0$ .

Geometric Sequences

Sequences with a constant ratio — AQA spec A24

A geometric sequence has a constant common ratio $r$ between consecutive terms: each term is obtained by multiplying the previous term by $r$ .

General form: $a,\; ar,\; ar^2,\; ar^3,\; \ldots$

nth term formula:

$T_n = ar^{n-1}$

Example: $3, 6, 12, 24, \ldots$ has $a = 3$ , $r = 2$ :

$T_n = 3 \times 2^{n-1}$

$T_6 = 3 \times 2^5 = 3 \times 32 = 96$

Finding the common ratio: $r = \dfrac{T_{k+1}}{T_k}$ (consistent for all consecutive pairs).

Behaviour:

$r$	Behaviour
$r > 1$	Diverges (grows without bound)
$0 < r < 1$	Converges to 0 (terms get smaller)
$r < 0$	Alternates in sign
$r = 1$	Constant sequence
$r = -1$	Alternates between $a$ and $-a$

AQA also expects you to recognise sequences of square numbers ( $1, 4, 9, 16, 25, \ldots$ i.e. $n^2$ ), cube numbers ( $1, 8, 27, 64, \ldots$ i.e. $n^3$ ) and triangular numbers ( $1, 3, 6, 10, 15, \ldots$ i.e. $\tfrac{1}{2}n(n+1)$ ).

Geometric sequences are not arithmetic — the difference between terms changes
If $r$ is a fraction (e.g. $r = 0.5$ ), the terms decrease but never reach zero
Triangular numbers are formed by adding consecutive integers: $1, 1+2, 1+2+3, \ldots$

Common pitfall

When asked to 'show a sequence is geometric', you must show that the ratio between all consecutive pairs is the same — checking just two consecutive terms is insufficient evidence.

Quadratic Sequences

Second differences and finding nth term of the form $an^2 + bn + c$ — AQA spec A25

A quadratic sequence has a constant second difference (the difference of the differences).

The nth term has the form:

$T_n = an^2 + bn + c$

Method to find $a$ , $b$ , $c$ :

Step 1: Write out the sequence and find first differences, then second differences.

$n$	1	2	3	4	5
$T_n$	3	8	15	24	35
1st diff		5	7	9	11
2nd diff			2	2	2

Step 2: $a = \dfrac{\text{second difference}}{2} = \dfrac{2}{2} = 1$ , so the $n^2$ term is $1 \cdot n^2$ .

Step 3: Subtract $an^2 = n^2$ from each term to leave a linear sequence:

$3 - 1 = 2,\; 8 - 4 = 4,\; 15 - 9 = 6,\; 24 - 16 = 8,\; 35 - 25 = 10$

Sequence: $2, 4, 6, 8, 10, \ldots$ — this has nth term $2n$ .

Step 4: Combine: $T_n = n^2 + 2n$ .

Check: $T_1 = 1 + 2 = 3$ ✓; $T_3 = 9 + 6 = 15$ ✓

If the constant $c \neq 0$ (e.g. $T_n = 2n^2 + 3n + 1$ ), the linear remainder sequence will have a non-zero constant — find its nth term as usual.

The second difference is always constant for a quadratic sequence; check this before proceeding
The coefficient of $n^2$ is half the second difference — never forget to halve
After subtracting $an^2$ , the remainder must be a linear (or constant) sequence; if it is not, recheck the second difference

Common pitfall

Forgetting to halve the second difference is the most common error: if the second difference is 4, then $a = 2$ , not 4. A quick way to spot this error: if $T_n = 4n^2 + \ldots$ has second difference 8 (not 4), the formula is wrong.

Special Sequences and Pattern Recognition

Triangular, square, cube numbers, Fibonacci and more — AQA spec A24

AQA expects you to recognise the following standard sequences on sight:

Sequence name	Terms	nth term
Square numbers	$1, 4, 9, 16, 25, \ldots$	$n^2$
Cube numbers	$1, 8, 27, 64, 125, \ldots$	$n^3$
Triangular numbers	$1, 3, 6, 10, 15, 21, \ldots$	$\dfrac{n(n+1)}{2}$
Powers of 2	$2, 4, 8, 16, 32, \ldots$	$2^n$
Powers of 10	$10, 100, 1000, \ldots$	$10^n$
Fibonacci-type	$1, 1, 2, 3, 5, 8, 13, \ldots$	$u_{n+2} = u_{n+1} + u_n$

Context/pattern questions: AQA often presents sequences as dot patterns or shapes and asks you to:

Find the next term (counting or spotting the rule)
Find the nth term (connect the pattern to position $n$ )
Decide if a large value (e.g. 500 tiles) is achievable

Strategy for pattern questions:

Build a table of position $n$ and term $T_n$
Check first and second differences
Form the nth term formula
Substitute the target value and check whether $n$ is a positive integer

Triangular numbers visualised as dot pyramids: each new row adds one more dot, giving 1, 3, 6, 10 for T₁ through T₄.

Triangular numbers appear in handshake problems, staircase patterns and Pascal's triangle diagonals
If a sequence involves $n^2$ , its second differences will be constant and equal to 2
The difference between consecutive square numbers is always an odd number: $4 - 1 = 3$ , $9 - 4 = 5$ , $16 - 9 = 7$ , $\ldots$ — this is the odd-number sequence

Common pitfall

On a pattern question, always build the table before writing down a formula. Students who guess 'it looks like $n^2$ ' without checking often get the wrong nth term when the sequence is $n^2 + 2$ or $2n^2 - 1$ .

Quick recap

A term-to-term rule requires the previous term; a position-to-term (nth term) formula works directly from $n$
nth term of an arithmetic sequence: $T_n = dn + (a - d)$ ; the coefficient of $n$ equals the common difference
nth term of a geometric sequence: $T_n = ar^{n-1}$ ; find $r$ by dividing any term by the previous
Special sequences: square ( $n^2$ ), cube ( $n^3$ ), triangular ( $\frac{n(n+1)}{2}$ ), Fibonacci (add previous two terms)
Quadratic sequence: constant second difference; $a = \frac{\text{second difference}}{2}$ ; subtract $an^2$ and find the linear remainder
To test if value $k$ is in the sequence, set $T_n = k$ and solve — $n$ must be a positive integer

Exam tips

Show your working for the differences table in full — difference rows are often awarded a method mark independently of the final nth term
Always verify your nth term by substituting $n = 1$ and $n = 2$ (or $n = 3$ ) before writing the final answer
For 'is $N$ a term of the sequence?' questions, solve the equation and explicitly state whether the answer is a positive integer — this is the conclusion the examiner is looking for
In context/pattern questions, write out the sequence as a table first — it anchors your working and prevents errors
Geometric sequences: if the ratio is a fraction, the terms decrease — ensure you can identify both increasing and decreasing geometric sequences
The word 'deduce' in an AQA question means your answer must follow from the algebra — show the derivation, not just the formula

Frequently asked questions

Sources: AQA GCSE Mathematics (8300) Specification (2026 onwards); AQA 8300 Examiners' Reports 2022–2024 · Last reviewed May 2026 · AQA GCSE · Mathematics 8300 Higher

Sequences

At a glance

What you should be able to do

Study notes

Term-to-Term and Position-to-Term Rules

Arithmetic (Linear) Sequences

Geometric Sequences

Quadratic Sequences

Special Sequences and Pattern Recognition

Quick recap

Exam tips

Frequently asked questions

How do I quickly find the nth term of a linear sequence?

What is the difference between an arithmetic and a geometric sequence?

How do I know if a sequence is quadratic rather than linear?

Can a sequence be both arithmetic and geometric?

How do I find which term has a particular value in a quadratic sequence?