IGCSE Mathematics: Sequences Practice Questions
For a linear sequence, the coefficient of n in the nth term is the common difference, and the constant is found by working back to the term before the first. Quadratic sequences have a constant second difference.
Topic 2.7 of Cambridge IGCSE Mathematics 0580 rewards a fixed method. Once you know that the first difference gives the coefficient of n and half the second difference gives the coefficient of n squared, every sequence question becomes mechanical. The questions below apply that.
What you need to know for Sequences
- Linear sequenceA sequence with a constant first difference. The nth term is dn + c, where d is the common difference and c is the value that would come before the first term.
- Quadratic sequenceA sequence with a constant second difference. The coefficient of n2 is half that second difference.
- Finding the rest of a quadratic nth termSubtract the n2 part from each term, leaving a linear sequence. Find its nth term and add it on.
- Geometric sequenceEach term is multiplied by a constant ratio r. The nth term is arn-1, where a is the first term.
- Which term equals a valueSet the nth term expression equal to the value and solve for n. A non-integer answer means the value is not in the sequence.
- Checking an nth termSubstitute n = 1, 2 and 3 and confirm you get the first three terms of the sequence.
IGCSE Mathematics Sequences questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Find the nth term of the sequence 5, 8, 11, 14, 17.
Show the worked answer
- Find the first differences: 8 minus 5 = 3, 11 minus 8 = 3, and so on. The common difference is 3.
- The sequence is linear, so the nth term starts with 3n.
- Compare 3n with the sequence: 3n gives 3, 6, 9, 12. Each actual term is 2 more.
- The nth term is 3n + 2. Check with n = 4: 3(4) + 2 = 14, which matches.
Find the nth term of the quadratic sequence 3, 8, 15, 24, 35.
Show the worked answer
- First differences: 5, 7, 9, 11. These are not constant, so the sequence is not linear.
- Second differences: 2, 2, 2. Constant, so the sequence is quadratic.
- The coefficient of n2 is half the second difference: 2 divided by 2 = 1, so the nth term begins with n2.
- Subtract n2 from each term: 3 minus 1 = 2, 8 minus 4 = 4, 15 minus 9 = 6, 24 minus 16 = 8. That linear sequence has nth term 2n, so the full nth term is n2 + 2n. Check with n = 5: 25 + 10 = 35.
A geometric sequence begins 2, 6, 18, 54. Write down an expression for the nth term and calculate the 6th term.
Show the worked answer
- Find the common ratio by dividing consecutive terms: 6 divided by 2 = 3, and 18 divided by 6 = 3. The ratio r is 3.
- The first term a is 2, so the nth term is a rn-1 = 2 x 3n-1.
- For the 6th term, substitute n = 6: 2 x 35.
- 35 = 243, so the 6th term is 2 x 243 = 486.
The nth term of a sequence is 4n minus 1. Determine whether 79 is a term of this sequence, and if so, which one.
Show the worked answer
- Set the nth term expression equal to the value: 4n minus 1 = 79.
- Add 1 to both sides: 4n = 80.
- Divide by 4: n = 20.
- n is a positive whole number, so 79 is a term of the sequence, and it is the 20th term. Had n come out as a fraction or a negative, 79 would not be in the sequence.
Common mistakes in this topic
- Using the first term as the constant in a linear nth term.
- Using the whole second difference rather than half of it for a quadratic.
- Writing a geometric nth term with index n rather than n minus 1.
- Confusing a term number with the value of the term.
- Not checking the nth term against the original sequence.
Exam tips
- Write the difference row underneath the sequence before doing anything else. It tells you immediately which type it is.
- Constant first difference means linear. Constant second difference means quadratic. Constant ratio means geometric.
- Always verify your nth term by substituting n equals 1, 2 and 3.
- For which term questions, form an equation and check that n is a positive whole number.
- For a quadratic, subtract the n squared part first and the remaining sequence is always linear.
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Sequences FAQs
How do I find the nth term of a linear sequence?
Find the common difference and use it as the coefficient of n. Then compare that multiple of n with the actual terms and add or subtract whatever constant makes them match. The constant equals the value that would come before the first term, not the first term itself.
How do I find the nth term of a quadratic sequence?
Find the second differences, which will be constant. Half of that second difference gives the coefficient of n squared. Subtract that part from every term, which leaves a linear sequence, find its nth term in the usual way, and add the two parts together.
What is the nth term of a geometric sequence?
It is a multiplied by r to the power n minus 1, where a is the first term and r is the common ratio found by dividing any term by the one before it. The index is n minus 1 so that substituting n equals 1 returns the first term.
How do I check whether a number is in a sequence?
Set the nth term expression equal to that number and solve for n. If n comes out as a positive whole number, the value is in the sequence and n tells you which term it is. A fractional or negative n means the value is not in the sequence.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.