What is a sequence in mathematics, and how is it relevant to the Edexcel IGCSE curriculum?

In mathematics, a sequence is an ordered list of numbers that follow a specific pattern or rule. Sequences are a fundamental concept in the Edexcel IGCSE curriculum, as they help students understand patterns, relationships, and functions, which are essential for solving complex mathematical problems.

How do you find the nth term of an arithmetic sequence in Edexcel IGCSE Mathematics?

To find the nth term of an arithmetic sequence, 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 allows students to determine any term in the sequence, which is a key skill in the Edexcel IGCSE Mathematics syllabus.

What is the difference between arithmetic and geometric sequences in the context of Edexcel IGCSE?

An arithmetic sequence is a sequence where each term is obtained by adding a constant value, known as the common difference, to the previous term. In contrast, a geometric sequence is one where each term is found by multiplying the previous term by a constant value called the common ratio. Understanding these differences is crucial for students studying Sequences in the Edexcel IGCSE Mathematics curriculum.

How can you determine if a sequence is arithmetic or geometric in Edexcel IGCSE Mathematics?

To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. For a geometric sequence, verify if the ratio between consecutive terms is constant. Recognizing these patterns is essential for solving sequence-related problems in the Edexcel IGCSE Mathematics course.

What are some common applications of sequences in real-world scenarios, as covered in the Edexcel IGCSE curriculum?

Sequences have various real-world applications, such as calculating interest in finance (geometric sequences) or predicting population growth. In the Edexcel IGCSE curriculum, students learn to apply sequences to solve practical problems, enhancing their understanding of mathematical concepts and their relevance to everyday life.

Why is "Using the WRONG coefficient of $n$" a common mistake in Sequences?

Why it happens: Confusing first term with common difference. How to avoid it: Common difference = coefficient of n. Then adjust constant. Always TEST n = 1.

Why is "Using FULL second difference as coefficient of $n^{2}$" a common mistake in Sequences?

Why it happens: Looks intuitive. How to avoid it: HALF the second difference = coefficient of n^2. So second difference = 2 → coef = 1. Source: 4MA1/2H May/Jun 2024 — examiner report Q14

Why is "Not verifying the $n$th term" a common mistake in Sequences?

Why it happens: Confidence. How to avoid it: ALWAYS check n = 1 and n = 2. Mark schemes credit verification.

Sequences, Functions and GraphsPearson Edexcel IGCSE Mathematics 4MA1 Higher

Sequences

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Sequences Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

Find the $n$ th term of arithmetic, geometric and quadratic sequences. Recognise patterns and produce general formulae.

What you’ll learn

Mapped to the Pearson Edexcel IGCSE 4MA1 syllabus (2026 onwards).

3.1 — Find the $n$ th term of an arithmetic sequence.
3.1 — Find the $n$ th term of a quadratic sequence.
3.1 — Identify and use geometric sequences.
3.1 — Use recursive (term-to-term) rules.

Arithmetic sequences

Constant first difference. $n$ th term: $a + (n-1)d$ .

Arithmetic = constant difference between consecutive terms.

Standard form: $u_{n} = a + (n - 1)d$ .

$a$ = first term.
$d$ = common difference.

Equivalent: $u_{n} = dn + (a - d)$ — useful for spotting the formula by inspection.

Example. $5, 8, 11, 14, \ldots$

The same amount is added each step — that constant is the common difference d.

$a = 5$ , $d = 3$ .
$u_{n} = 5 + (n - 1)(3) = 3n + 2$ .
Verify: $n = 1$ : $5$ ✓. $n = 2$ : $8$ ✓.

Spotting by inspection (Edexcel preferred method).

Common difference = coefficient of $n$ .
Adjust constant: subtract that coefficient from the first term to get the constant.
$5, 8, 11, \ldots$ : $d = 3$ , $5 - 3 = 2$ . So $u_{n} = 3n + 2$ .

Worked example. Find the $n$ th term of $40, 35, 30, 25, \ldots$

$d = -5$ .
$40 - (-5) = 45$ .
$u_{n} = -5n + 45$ or $45 - 5n$ .

Edexcel tip. Verify by substituting $n = 1$ and $n = 2$ . Mark scheme credits this verification.

Constant first difference $d$ .
$u_{n} = a + (n-1)d$ .
By inspection: $d \cdot n +$ constant.
Verify $n = 1, 2$ .

Quadratic sequences

Constant SECOND difference. $n^{2}$ coefficient = half the second difference.

Quadratic sequence = constant SECOND differences (the differences between the differences).

$n$ th term form: $u_{n} = An^{2} + Bn + C$ .

Method:

Find first differences.
Find second differences. Should be constant.
$A =$ half the second difference.
Subtract $An^{2}$ from each term — the result is arithmetic.
Find $B$ (and $C$ ) from the arithmetic part.

Example. $5, 11, 19, 29, 41, \ldots$

Constant second difference 2 confirms a quadratic; the n² coefficient is half of it.

First differences: $6, 8, 10, 12$ .
Second differences: $2, 2, 2$ . ✓ Quadratic.
$A = 2/2 = 1$ . So $n^{2}$ part is $n^{2}$ .
Subtract $n^{2}$ from sequence: $5 - 1 = 4$ , $11 - 4 = 7$ , $19 - 9 = 10$ , $29 - 16 = 13$ , $41 - 25 = 16$ .
$4, 7, 10, 13, 16$ is arithmetic with $d = 3$ , first term $4$ .
$u_{n} = 3n + 1$ for the linear part.
Combined: $u_{n} = n^{2} + 3n + 1$ .

Verify. $n = 1$ : $1 + 3 + 1 = 5$ ✓. $n = 5$ : $25 + 15 + 1 = 41$ ✓.

Worked qualitative. Why does the $n^{2}$ coefficient equal HALF the second difference?

For $u_{n} = An^{2} + Bn + C$ : first differences are $A(2n - 1) + B$ — linear in $n$ .
Second differences are constant $= 2A$ .
So $A =$ half.

Edexcel tip. Always state the second difference and apply the half rule. Mark schemes give M1 for the second difference, M1 for $A$ .

Second difference constant.
$A =$ half the second diff.
Subtract $An^{2}$ , fit arithmetic.
$u_{n} = An^{2} + Bn + C$ .

Geometric sequences

Constant ratio. $u_{n} = ar^{n-1}$ .

Geometric = constant RATIO between consecutive terms.

Form: $u_{n} = ar^{n-1}$ .

$a$ = first term.
$r$ = common ratio.

Examples:

Sequence	$a$	$r$	$u_{n}$
$2, 6, 18, 54, \ldots$	$2$	$3$	$2 \cdot 3^{n-1}$
$4, 8, 16, 32, \ldots$	$4$	$2$	$4 \cdot 2^{n-1}$
$100, 50, 25, 12.5, \ldots$	$100$	$1/2$	$100 \cdot (1/2)^{n-1}$
$3, -6, 12, -24, \ldots$	$3$	$-2$	$3 \cdot (-2)^{n-1}$

Each term is multiplied by the same ratio r — so the sequence grows multiplicatively.

Worked qualitative. What's the difference between $r > 1$ and $r < 1$ ?

$r > 1$ : terms grow (compound interest, exponential growth).
$0 < r < 1$ : terms shrink (depreciation, decay).
$r$ negative: terms alternate signs.

Edexcel tip. State $a$ and $r$ explicitly. Mark schemes credit each.

$u_{n} = ar^{n-1}$ .
$r$ = common ratio.
$r > 1$ : grow. $0 < r < 1$ : shrink.
Negative $r$ : alternates signs.

Recursive sequences

Each term defined in terms of previous one(s).

Recursive (term-to-term) rule defines each term using earlier terms.

Standard notation: $u_{1} =$ first term. $u_{n} =$ formula in $u_{n-1}$ (and possibly $u_{n-2}$ ).

Examples:

Arithmetic recursive: $u_{1} = 5$ , $u_{n} = u_{n-1} + 3$ . Equivalent to common difference $3$ .

Geometric recursive: $u_{1} = 2$ , $u_{n} = 3 u_{n-1}$ . Equivalent to common ratio $3$ .

Fibonacci-like: $u_{1} = 1, u_{2} = 1, u_{n} = u_{n-1} + u_{n-2}$ . Each term is the sum of two previous.

Worked example. $u_{1} = 2$ , $u_{2} = 5$ , $u_{n} = u_{n-1} + u_{n-2}$ . Find $u_{5}$ .

$u_{3} = 5 + 2 = 7$ .
$u_{4} = 7 + 5 = 12$ .
$u_{5} = 12 + 7 = 19$ .

Worked qualitative. Why are recursive sequences harder to find general $n$ th terms for?

Each step depends on previous — no closed form in general.
Some have closed forms (Fibonacci has one — too complex for IGCSE).
Edexcel only expects you to GENERATE terms, not find a closed formula.

Edexcel tip. Recursive questions ask you to compute terms one at a time. Show each step.

$u_{n} =$ formula in $u_{n-1}$ .
Generate term-by-term.
Common form: arithmetic, geometric, Fibonacci-style.
No closed formula expected at IGCSE.

How it’s examined

Sequences appear every Paper 1H + 2H (3-7 marks). Arithmetic is foundation; quadratic for higher candidates. Examiner reports flag (1) using full second difference instead of half, (2) wrong sign in arithmetic constant, (3) not verifying.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/1H Jan 2024, 4MA1/2H May/Jun 2024. Last reviewed 2026-05-07.

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Step-by-step worked examples — Sequences

Step-by-step solutions to past-paper-style questions on sequences, written exactly the way a tutor would explain them at the board.

1 $n$ th term of an arithmetic sequence
Foundation• arithmetic, nth term
Question
Find the $n$ th term of $5, 8, 11, 14, \ldots$
Step-by-step solution
1. Step 1
  Common difference: $8 - 5 = 3$ . So coefficient of $n$ is $3$ .
2. Step 2
  Test: $3n$ gives $3, 6, 9, 12, \ldots$ . Adjust by adding the difference between $5$ (first term) and $3$ (first $3n$ ): $5 - 3 = 2$ .
3. Step 3
  $n$ th term: $3n + 2$ .
4. Step 4
  Verify: $n = 1$ : $5$ ✓. $n = 4$ : $14$ ✓.
Answer
$3n + 2$
Examiner tip
Always VERIFY by substituting $n = 1$ and $n = 2$ . Mark schemes credit checking.
2Use the $n$ th term to find a specific term
Foundation• Adapted from 4MA1/1H Jan 2024 Q5• nth term
Question
The $n$ th term of a sequence is $4n - 7$ . Find the 20th term and check whether $73$ is in the sequence.
Step-by-step solution
1. Step 1
  20th term: $4(20) - 7 = 73$ .
2. Step 2
  Set $73 = 4n - 7$ : $4n = 80$ , $n = 20$ . Yes — the 20th term is $73$ .
Answer
20th term = $73$ . $73$ IS in the sequence (term $20$ ).
3 $n$ th term of a quadratic sequence
Higher• Adapted from 4MA1/2H May/Jun 2024 Q14• quadratic sequence
Question
Find the $n$ th term of $5, 11, 19, 29, 41, \ldots$
Step-by-step solution
1. Step 1
  First differences: $6, 8, 10, 12$ .
2. Step 2
  Second differences: $2, 2, 2$ — constant. Confirms quadratic.
3. Step 3
  Coefficient of $n^{2}$ : half of second difference = $1$ . So formula starts with $n^{2}$ .
4. Step 4
  Subtract $n^{2}$ from sequence: $5 - 1 = 4, 11 - 4 = 7, 19 - 9 = 10, 29 - 16 = 13, 41 - 25 = 16$ . Now $4, 7, 10, 13, 16$ is arithmetic with difference $3$ .
5. Step 5
  $n$ th term of this arithmetic: $3n + 1$ .
6. Step 6
  Combined: $n^{2} + 3n + 1$ .
Answer
$n^{2} + 3n + 1$
Examiner tip
Method: use second differences. Half of second difference = coefficient of $n^{2}$ . Then subtract the $n^{2}$ values, treating remainder as arithmetic.
4Geometric sequence
Higher• geometric
Question
A geometric sequence has first term $3$ and common ratio $2$ . Find the $n$ th term and the 6th term.
Step-by-step solution
1. Step 1
  Geometric: $a, ar, ar^{2}, ar^{3}, \ldots$ . So $n$ th term: $ar^{n-1}$ .
2. Step 2
  Substitute: $a = 3$ , $r = 2$ . $n$ th term = $3 \cdot 2^{n-1}$ .
3. Step 3
  6th term: $3 \cdot 2^{5} = 3 \cdot 32 = 96$ .
Answer
$n$ th term: $3 \cdot 2^{n-1}$ . 6th term: $96$ .
5Recursive (Fibonacci-style) sequence
Higher• recursive
Question
A sequence is given by $u_{1} = 2$ , $u_{2} = 5$ , and $u_{n} = u_{n-1} + u_{n-2}$ . Find $u_{5}$ .
Step-by-step solution
1. Step 1
  $u_{3} = u_{2} + u_{1} = 5 + 2 = 7$ .
2. Step 2
  $u_{4} = u_{3} + u_{2} = 7 + 5 = 12$ .
3. Step 3
  $u_{5} = u_{4} + u_{3} = 12 + 7 = 19$ .
Answer
$u_{5} = 19$

Key Formulae — Sequences

The formulae you need to memorise for sequences on the Pearson Edexcel IGCSE 4MA1 paper, with every variable defined in plain English and a note on when to use it.

Arithmetic sequence $n$ th term
$u_{n} = a + (n - 1)d$
$a$
first term
$d$
common difference
$n$
term number
When to use
Linear sequences with constant first difference.
Example
$a = 5, d = 3$ : $u_{n} = 5 + (n-1)(3) = 3n + 2$ .
Geometric sequence $n$ th term
$u_{n} = ar^{n - 1}$
$a$
first term
$r$
common ratio
When to use
Sequences where each term is multiplied by a constant ratio.
Example
$a = 3, r = 2$ : $u_{n} = 3 \cdot 2^{n-1}$ . So $u_{6} = 3 \cdot 32 = 96$ .

Key Definitions and Keywords — Sequences

Definitions to memorise and the exact keywords mark schemes credit for sequences answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IGCSE 4MA1 sitting.

Arithmetic sequence
Examiner keyword
A sequence with a CONSTANT first difference (common difference $d$ ).
Example
$3, 7, 11, 15, \ldots$ has $d = 4$ .
Geometric sequence
Examiner keyword
A sequence with a CONSTANT ratio between consecutive terms.
Example
$2, 6, 18, 54, \ldots$ has $r = 3$ .
Quadratic sequence
Examiner keyword
A sequence whose SECOND differences are constant. $n$ th term has $n^{2}$ .
Example
$5, 11, 19, 29, \ldots$ — second difference $= 2$ .
$n$ th term (general term)
A formula in $n$ giving the $n$ th term of a sequence directly.

Common Mistakes and Misconceptions — Sequences

The traps other students keep falling into on sequences questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕Using the WRONG coefficient of $n$
Why it happens
Confusing first term with common difference.
How to avoid it
Common difference = coefficient of $n$ . Then adjust constant. Always TEST $n = 1$ .
✕Using FULL second difference as coefficient of $n^{2}$
4MA1/2H May/Jun 2024 — examiner report Q14
Why it happens
Looks intuitive.
How to avoid it
HALF the second difference = coefficient of $n^{2}$ . So second difference $= 2$ → coef = $1$ .
✕Not verifying the $n$ th term
Why it happens
Confidence.
How to avoid it
ALWAYS check $n = 1$ and $n = 2$ . Mark schemes credit verification.

Practice questions

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Sequences — frequently asked questions

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Sequences

Short Study Notes in the form of Slides

Short Study Notes — Sequences

Detailed Notes

What you’ll learn

How it’s examined

1nnnth term of an arithmetic sequence

2Use the nnnth term to find a specific term

3nnnth term of a quadratic sequence

4Geometric sequence

5Recursive (Fibonacci-style) sequence

Arithmetic sequence nnnth term

Geometric sequence nnnth term

Arithmetic sequence

Geometric sequence

Quadratic sequence

nnnth term (general term)

✕Using the WRONG coefficient of nnn

✕Using FULL second difference as coefficient of n2n^{2}n2

✕Not verifying the nnnth term

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Sequences — frequently asked questions

1 $n$ th term of an arithmetic sequence

2Use the $n$ th term to find a specific term

3 $n$ th term of a quadratic sequence

Arithmetic sequence $n$ th term

Geometric sequence $n$ th term

$n$ th term (general term)

✕Using the WRONG coefficient of $n$

✕Using FULL second difference as coefficient of $n^{2}$

✕Not verifying the $n$ th term