Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want sequences and nth-term rules — arithmetic progressions, quadratic sequences and position-to-term patterns — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise sequences and the nth term in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the sequences revision-guide angle, while Tutopiya’s Sequences and nth Term subtopic page owns the learning resource and the free sequences quiz owns the practice.

Sequences and nth-term formulae appear regularly in Cambridge IGCSE Mathematics (0580/0607) Algebra papers. Examiners test whether you can move from a list of terms to a rule, and from a rule to any term number — often in questions worth two to four marks. This guide explains the subtopic, decodes the command words you will meet, and points you to focused practice.

Key takeaways

• An arithmetic sequence has a constant first difference; its nth term is a + (n − 1)d.
• A quadratic sequence has a constant second difference; the nth term has the form an² + bn + c.
• “nth term” means a formula in n that generates the sequence from n = 1, 2, 3, …
• Always substitute n = 1 to check your formula matches the first term.

What are sequences and the nth term in Cambridge IGCSE Maths?

A sequence is an ordered list of numbers following a rule. The nth term is a formula that gives the value of the term in position n. In Cambridge IGCSE Mathematics you must find nth-term rules for arithmetic sequences, quadratic sequences and simple special patterns, then use the formula to find particular terms or term numbers.

Read the worked examples on Tutopiya’s Sequences and nth Term subtopic page before attempting questions.

The core ideas you must master

Idea	What it means	How the exam uses it
Term / position	The kth term is the number in position k	”Write down the 10th term”
First difference	Subtract consecutive terms	Signals an arithmetic sequence
Second difference	Difference of first differences	Signals a quadratic nth term
nth term (arithmetic)	Tₙ = a + (n − 1)d	”Find the nth term of 5, 8, 11, 14, …“
nth term (quadratic)	Tₙ = an² + bn + c	”Find the nth term of 2, 8, 18, 32, …”

How to find the nth term of an arithmetic sequence — step by step

1. Write the terms and find the first difference d between consecutive terms.
2. If d is constant, the sequence is arithmetic. The first term is a.
3. Write Tₙ = a + (n − 1)d and simplify if asked.
4. Check by substituting n = 1, 2, 3.
5. To find which term equals a value, set Tₙ equal to that value and solve for n.

Test yourself with the free Sequences quiz.

How to find the nth term of a quadratic sequence

1. Find the first difference, then the second difference.
2. If the second difference is constant, divide it by 2 to get a in an² + bn + c.
3. Build a table: compare an² with the original terms to find what to add (bn + c).
4. Alternatively, solve for a, b, c using the first three terms.
5. Check with n = 1, 2, 3, 4.

Sequences in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Write down the next term	Continue the pattern — no formula needed	”Write down the next term in the sequence 3, 7, 11, 15, …”
Find the nth term	Formula in n	”Find the nth term of the sequence 4, 7, 10, 13, …”
Find the value of the nth term when n = …	Substitute into your formula	”Find the 20th term of the sequence.”
Find which term has value …	Solve Tₙ = k for n	”Find n if the nth term is 52.”
Show that	Prove a given nth-term rule	”Show that the nth term of … is 3n + 2.”

Worked exam-style stems (how to answer the wording)

1. “Find the nth term of the sequence 5, 9, 13, 17, …” First difference 4 → arithmetic. a = 5, d = 4 → Tₙ = 5 + 4(n − 1) = 4n + 1. Reward: correct difference, simplified formula.
2. “The nth term of a sequence is 2n + 3. Find the 15th term.” Substitute n = 15 → 2(15) + 3 = 33. Reward: correct substitution.
3. “Find the nth term of the sequence 1, 4, 9, 16, …” Second differences constant → quadratic. Pattern is square numbers: Tₙ = n². Reward: recognising n² or full quadratic method.

Work the full set on the Algebra topical past-paper questions.

How sequences connect to the rest of Algebra

Arithmetic sequences link to linear expressions in Linear Equations and Inequalities; quadratic sequences reuse skills from Quadratic Equations. The Cambridge IGCSE Maths resource hub helps you move between subtopics efficiently.

Common mistakes students make

• Using n as the first term instead of the position (n = 1 for the first term).
• Forgetting to simplify a + (n − 1)d into the form an + b.
• Treating a quadratic sequence as arithmetic because the first difference is not constant enough at a glance.
• Finding the nth term but not checking with n = 1.
• When solving Tₙ = k, getting a non-integer n and not realising the value is not a term in the sequence.

When you need more support

If nth-term questions keep slipping — especially quadratic sequences — take the Sequences quiz again, work through the Algebra topical past-paper questions, and ask a Cambridge IGCSE Maths tutor to fix the gap.

Frequently asked questions

What is the nth term formula for an arithmetic sequence? Tₙ = a + (n − 1)d, where a is the first term and d is the common difference between consecutive terms.

How do I know if a sequence is quadratic? The second differences (differences of the first differences) are constant. The nth term then has the form an² + bn + c.

Can the nth term be a fraction for some n? The formula is defined for all positive integers n, but if you solve Tₙ = k and get a non-integer, that value k is not a term in the sequence.

How do I revise sequences effectively? Master arithmetic nth terms first, then quadratic patterns. Use the sequences quiz after each stage to confirm the method.

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