Find the sum of the first 25 terms of $5+8+11+\cdots$.

$a=5,\ d=3,\ n=25$. $S_n=\tfrac{n}{2}(2a+(n-1)d)=\tfrac{25}{2}(10+72)$. $=1025$. Answer: 1025

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Edexcel IGCSE Maths (4MA1) · Unit 3: Equations & Sequences

Edexcel IGCSE 4MA1 Sequences: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

A linear (arithmetic) sequence goes up or down by the same amount each time: the common difference $d$ . Its nth term is $a + (n - 1) d$ , where $a$ is the first term, which simplifies to $d n + (a - d)$ . The sum of the first $n$ terms is $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ , which is on the 4MA1 Higher formulae sheet.

Unit 3: Equations & Sequences
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Notes
Formulas
Worked examples
Mistakes
FAQ

Sequences notes

Term-to-term and nth-term rules

A term-to-term rule says how to get from one term to the next, such as "add 4". An nth-term rule gives any term from its position $n$ : for $3 n + 2$ , the 10th term is $32$ .

The nth term of a linear sequence

Find the common difference, then compare with that times table. 5, 9, 13, 17, … goes up by 4, so start with $4 n$ (4, 8, 12, 16); each term is 1 more, so the nth term is $4 n + 1$ . The formula $a + (n - 1) d$ gives the same: $5 + 4 (n - 1) = 4 n + 1$ .

Is a number in the sequence?

Set the nth term equal to the number and solve. $4 n + 1 = 150$ gives $n = 37.25$ , which is not a whole number, so 150 is not a term. $4 n + 1 = 149$ gives $n = 37$ , so 149 is the 37th term.

Arithmetic series

Use $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ . The sum of the first 20 terms of 5, 9, 13, … is $\frac{20}{2} [10 + 19 \times 4] = 860$ . Questions can also give you the sum and ask for $n$ , $a$ or $d$ : substitute what you know and solve.

Other sequences

Not every sequence is linear. The square numbers 1, 4, 9, 16, … have nth term $n^{2}$ , and 2, 6, 18, 54, … multiplies by 3 each time. Check the differences first: if they are not all the same, the sequence is not linear.

Key formulas

Arithmetic $n$ th term Learn it: $u_{n} = a + (n - 1) d$
Arithmetic series, sum of $n$ terms On the formula sheet: $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

On the formula sheet = printed on the formulae sheet in the 4MA1 Higher papers, so know where to find it. Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Sequences worked examples

Worked example 1: nth term of a linear sequence

Find the nth term of 2, 9, 16, 23, …

Solution

Common difference 7, so start with $7 n$ : 7, 14, 21, 28.
Each term is 5 less, so the nth term is $7 n - 5$ .

3 steps

Tip: Check: n = 1 gives 2.

Worked example 2: nth term of an arithmetic sequence

An arithmetic sequence has first term 7 and common difference 4. Find the 20th term.

Solution

$u_{n} = a + (n - 1) d$ .
$u_{20} = 7 + 19 \times 4 = 83$ .

3 steps

Tip: There are 19 steps from the 1st to the 20th term.

Worked example 3: Sum of an arithmetic series

Find the sum of the first 25 terms of $5 + 8 + 11 + \dots$ .

Solution

$a = 5, d = 3, n = 25$ .
$S_{n} = \frac{n}{2} (2 a + (n - 1) d) = \frac{25}{2} (10 + 72)$ .
$= 1025$ .

4 steps

Tip: Use n − 1, not n, with d.

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Sequences exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

Here are the first four terms of a sequence: 7, 11, 15, 19. (a) Write down the next two terms. (b) Write down the term-to-term rule.

Show the answer and mark scheme

B1 23, 27
B1 add 4

Answer: (a) 23, 27 (b) add 4

Question 2 · easy · 2 marks

Find an expression for the nth term of the sequence 5, 12, 19, 26, …

Show the answer and mark scheme

M1 7n + k for any k, or common difference 7 identified
A1 7n − 2

Answer: $7 n - 2$

Question 3 · easy · 2 marks

Find an expression for the nth term of the sequence 50, 46, 42, 38, …

Show the answer and mark scheme

M1 −4n + k for any k
A1 54 − 4n oe

Answer: $54 - 4 n$

Question 4 · medium · 2 marks

The nth term of a sequence is $3 n + 8$ . Is 100 a term of the sequence? Give a reason for your answer.

Full worked solution Free, no sign-in

M1 3n + 8 = 100 or n = 92/3 oe, or terms 98 and 101 shown
A1 No, with correct reason (n is not a whole number)

Answer: No: $3 n + 8 = 100$ gives $n = 30 \frac{2}{3}$ , not an integer

Every question here has a full worked solution. See them all →

Question 5 · medium · 2 marks

The nth term of a sequence is $6 n - 5$ . Find the first term of the sequence that is greater than 200.

Show the answer and mark scheme

M1 6n − 5 > 200 or n > 34.1…
A1 205

Answer: 205

Question 6 · medium · 2 marks

The first term of a sequence is 5. The term-to-term rule is 'multiply by 2 then subtract 3'. Find the 4th term of the sequence.

Show the answer and mark scheme

M1 second term 7 or third term 11
A1 19

Answer: 19

Question 7 · hard · 3 marks

Sequence A has nth term $4 n + 1$ . Sequence B has nth term $61 - 2 n$ . For one value of $n$ the nth terms of the two sequences are equal. Find this value of $n$ and the value of that term.

Show the answer and mark scheme

M1 4n + 1 = 61 − 2n
A1 n = 10
A1 41

Answer: $n = 10$ , term 41

Question 8 · hard · 3 marks

Pattern 1 is made from 6 matchsticks, pattern 2 from 11 matchsticks and pattern 3 from 16 matchsticks. The patterns continue in the same way. (a) How many matchsticks are in pattern 40? (b) Which pattern uses exactly 136 matchsticks?

Show the answer and mark scheme

M1 nth term 5n + 1 or 5 × 40 + 1
A1 201
B1 27

Answer: (a) 201 (b) pattern 27

Common sequences mistakes

Using n instead of n − 1. The 20th term is $a + 19 d$ : there are 19 steps from the 1st term to the 20th.
Giving the term-to-term rule as the nth term. "Add 4" is not an nth term. The nth term of 5, 9, 13, … is $4 n + 1$ , not $n + 4$ .
The wrong constant. Check your nth term with n = 1: it must give the first term.
A decimal n. If solving nth term = a number gives a decimal n, the number is not in the sequence. Say so, and give that reason.

Where students lose marks across 4MA1, from the examiner reports →

Sequences FAQ

Is the arithmetic series formula on the 4MA1 formula sheet?

Yes. The sum of the first n terms of an arithmetic series is printed on the 4MA1 Higher formulae sheet. The nth-term formula a + (n − 1)d is not, so learn that one.

How do I find the nth term of a decreasing sequence?

The common difference is negative. 20, 17, 14, 11, … goes down by 3, so start with −3n, which gives −3 when n = 1. The first term is 20, so add 23: the nth term is 23 − 3n.

Can I use a calculator for sequences in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Sequences revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 sequences questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Sequences: exam questions and answers, notes and worked examples

Try it first

Sequences notes

Term-to-term and nth-term rules

The nth term of a linear sequence

Is a number in the sequence?

Arithmetic series

Other sequences

Key formulas

Sequences worked examples

Worked example 1: nth term of a linear sequence

Solution

Worked example 2: nth term of an arithmetic sequence

Solution

Worked example 3: Sum of an arithmetic series

Solution

Sequences exam-style questions

Question 1 · easy · 2 marks

Question 2 · easy · 2 marks

Question 3 · easy · 2 marks

Question 4 · medium · 2 marks

Full worked solution Free, no sign-in

Question 5 · medium · 2 marks

Question 6 · medium · 2 marks

Question 7 · hard · 3 marks

Question 8 · hard · 3 marks

Common sequences mistakes

Sequences FAQ

Next steps

Practise sequences with your progress saved

More Unit 3 topics: Equations & Sequences