Sequences

Q: Do I need to memorise the arithmetic nth term formula for Higher Tier?

Yes, the nth term formula is not provided on the formula sheet, so you must recall it. The sum of n terms formula is provided for Higher Tier.

Edexcel International GCSE Mathematics A· 3.1· 25 min read

1. Term-to-Term and Position-to-Term Sequence Rules★☆☆☆☆⏱ 10 min

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Sequences are ordered lists of numbers (called terms) that follow a consistent rule. The two most common ways to define sequences are term-to-term rules and position-to-term rules. Term-to-term rules describe how to move from one term to the next, while position-to-term rules calculate any term using its position n in the sequence.

📘 Definition

Common Sequence Types

Standard sequences you will encounter include odd numbers, even numbers, square numbers, multiples of a number, and powers (e.g. powers of 2).

📐 Worked Example

A sequence follows the term-to-term rule: multiply by 2 and subtract 3. The first term is 4. Write the first 4 terms of the sequence.

1
1st term (n=1): given as 4
2
2nd term: $(4 \times 2) - 3 = 8 - 3 = 5$
3
3rd term: $(5 \times 2) - 3 = 10 - 3 = 7$
4
4th term: $(7 \times 2) - 3 = 14 - 3 = 11$

✓ Quick check

A sequence has position-to-term rule $3 n + 2$ . What is the 5th term?
- 15
- 17
- 19
- 21
Reveal answer
17 —
Substitute n=5: $3 (5) + 2 = 15 + 2 = 17$ .

Exam tip:

Always check 2-3 terms after deriving a rule to confirm it matches the given sequence.

2. Linear nth Term of Arithmetic Sequences (Foundation & Higher)★★☆☆☆⏱ 12 min

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Arithmetic sequences are sequences where the difference between consecutive terms is constant (called the common difference). For Foundation Tier, you need to write the linear nth term expression for these sequences, which takes the form $d n + c$ , where d is the common difference and c is a constant.

📘 Definition

Linear nth Term Rule (Foundation)

To find the nth term: 1. Calculate the common difference d between consecutive terms. 2. Write $d n$ , then substitute n=1 to find the constant c needed to match the first term.

📐 Worked Example

Find the nth term of the sequence 5, 9, 13, 17, ...

1
Calculate the common difference d: $9 - 5 = 4$ , $13 - 9 = 4$ , so $d = 4$ .
2
Start with the expression $4 n$ . Substitute n=1: $4 (1) = 4$ .
3
The first term of the sequence is 5, so we need to add 1 to 4 to get 5: $c = 1$ .
4
Final nth term: $4 n + 1$

Exam tip:

If the sequence is decreasing, your common difference d will be negative, e.g. for 10, 7, 4, 1..., $d = - 3$ .

3. Arithmetic Sequences: a and d Formulation (Higher Only)★★★☆☆Higher only⏱ 10 min

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For Higher Tier, you will use the formal notation for arithmetic sequences: $a$ = first term, $d$ = common difference. The nth term formula is $u_{n} = a + (n - 1) d$ , which you must memorise as it is not provided on the formula sheet. You will often be asked to find a and d given two terms of the sequence.

📘 Definition

nth Term Formula (Higher Only)

$u_{n} = a + (n - 1) d$

Calculates the value of the nth term of an arithmetic sequence, where $a$ = first term, $d$ = common difference, $n$ = position of term.

📐 Worked Example

The 2nd term of an arithmetic sequence is 7, and the 5th term is 19. Find the values of a and d.

1
Write equations for the given terms using the nth term formula:
2
$u_2 = a + (2-1)d = a + d = 7$
3
$u_5 = a + (5-1)d = a + 4d = 19$
4
Subtract the first equation from the second to eliminate a: $(a + 4 d) - (a + d) = 19 - 7$ → $3 d = 12$ → $d = 4$
5
Substitute d=4 back into the first equation: $a + 4 = 7$ → $a = 3$
6
Check with the 5th term: $3 + 4 (4) = 19$ , which matches the given value.

4. Sum of an Arithmetic Series (Higher Only)★★★★☆Higher only⏱ 15 min

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An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first n terms $S_{n}$ is provided on the Higher Tier formula sheet: $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ . You can use this formula directly once you know a, d and n.

📐 Worked Example

Find the sum of the first 50 terms of the arithmetic series: 4 + 7 + 10 + 13 + ...

1
Identify the required values: first term $a = 4$ , common difference $d = 7 - 4 = 3$ , number of terms $n = 50$ .
2
Substitute into the sum formula:
3
$S_{50} = \frac{50}{2}[2(4) + (50-1)(3)]$
4
$= 25[8 + 49(3)] = 25[8 + 147] = 25[155]$
5
$= 3875$

Exam tip:

If you are asked for the sum of terms between two positions (e.g. sum from term 10 to term 20), calculate $S_{20} - S_{9}$ to get the correct total.

5. Common Pitfalls

Wrong move:

Using $a + n d$ instead of $a + (n - 1) d$ for the nth term (Higher Tier)

Why:

The first term is n=1, so the difference is added (n-1) times, not n times, leading to off-by-one errors.

Correct move:

Memorise the formula $u_{n} = a + (n - 1) d$ and test it against the first term to confirm it gives the right value when n=1.

Wrong move:

Confusing term-to-term and position-to-term rules

Why:

If a question asks for the nth term, a term-to-term rule (e.g. add 4) will get zero marks, as you need to give a formula in terms of n.

Correct move:

Check the question wording carefully: if it asks for the nth term or position-to-term rule, give an expression with n, not a recurrence rule.

Wrong move:

Using a positive d for decreasing sequences

Why:

If the sequence is getting smaller each time, the common difference is negative, so your nth term will be incorrect if you use a positive d.

Correct move:

Always calculate d as next term minus previous term, even if it gives a negative value, and test your nth term against the second term to confirm.

Wrong move:

Calculating $S_{20} - S_{10}$ for the sum of terms from 10 to 20

Why:

Subtracting S_10 removes the 10th term, so you miss it from your total sum.

Correct move:

Subtract $S_{n - 1}$ from $S_{m}$ when calculating the sum between term n and term m, e.g. $S_{20} - S_{9}$ for terms 10 to 20.

Wrong move:

Trying to use geometric sequence rules for non-arithmetic sequences

Why:

Geometric sequences are out of scope for Edexcel IGCSE Maths A (4MA1), so all sequence questions will be arithmetic or use simple term-to-term rules.

Correct move:

Only use arithmetic sequence formulae for sequences with constant difference; for other sequences, use the given term-to-term rule directly.

6. Quick Reference Cheatsheet

Concept	Foundation Tier	Higher Tier
Term-to-term rule	Describes how to get next term from previous (e.g. add 4)	Same as Foundation
Position-to-term rule	Linear expression for nth term: $d n + c$	Memorised formula: $u_{n} = a + (n - 1) d$
Find a and d	Not required	Solve simultaneous equations from given terms
Sum of first n terms	Not required	Use provided formula: $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

7. Frequently Asked

Do I need to memorise the arithmetic nth term formula for Higher Tier?

Yes, the nth term formula $a + (n - 1) d$ is not provided on the formula sheet, so you must recall it. The sum of n terms formula $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ is provided for Higher Tier.

What is the difference between term-to-term and position-to-term rules?

A term-to-term rule tells you how to get the next term if you know the previous one (e.g. add 4 each time). A position-to-term (nth term) rule lets you calculate any term directly using its position number n, without working through all previous terms.

Going deeper

formula_sheetEdexcel IGCSE Maths A Higher Formula SheetContains arithmetic series sum formula for Higher Tier

What's Next

Now that you have mastered sequences for Edexcel IGCSE Maths A (4MA1), you are ready to progress to related topics in the Sequences, Functions and Graphs unit. Next, you will learn how to plot and interpret linear graphs, which builds directly on your understanding of linear nth term expressions. For Higher Tier students, this sequence knowledge will also be used to solve problems involving algebraic patterns and real-world applications that follow arithmetic rules. Make sure to practice past paper questions for sequences to familiarise yourself with common exam phrasing and avoid the common pitfalls outlined in this guide, as this topic appears frequently in both Foundation and Higher papers.