Continue sequences and find general terms
Use term-to-term rules and differences to derive linear, quadratic and higher-order sequence patterns.
Section 2 · Algebra
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the difference in 3, 8, 13, 18, …?
What you will learn
- Continue a sequence and describe its term-to-term rule
- Find nth terms for linear sequences
- Use differences for quadratic and cubic patterns
- Recognise exponential sequences and combinations for Extended
A general term gives any position
The term-to-term rule moves one place at a time; the nth term gives the value at position n. Unless told otherwise, the first term uses n = 1.
For a linear sequence with constant difference d, start with dn and adjust so n = 1 gives the first term. Check the rule against more than one term.
Difference tables identify higher-order patterns
A quadratic sequence has constant second differences; a cubic has constant third differences. An exponential sequence has a constant ratio rather than a constant difference.
Extended includes linear, quadratic, cubic, exponential sequences and combinations. A short list may fit more than one rule, so follow the intended pattern or contextual evidence.
Pause and explain
Find the nth term of 4, 7, 10, 13, …
Worked example
Find the nth term of 5, 9, 13, 17, … and describe 2, 6, 12, 20, …
Show the worked solution
- First differences are 4, so start with 4n.
- At n = 1 the term must be 5, so add 1.
- The second list follows n(n + 1).
Answer 4n + 1; n(n + 1) = n² + n.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the nth term of 6, 10, 14, 18, … and next two terms of 1, 4, 9, 16, …
Give me a hint
Use the constant difference for the first and square numbers for the second.
Compare my reasoning
- The difference is 4, so begin with 4n.
- The first term is 6, so add 2: 4n + 2.
- The next squares are 25 and 36.
4n + 2; 25 and 36.
Look for these in your work
- Difference found
- Constant checked at n = 1
- Square pattern extended
Find differences and nth term for 2, 6, 12, 20, 30.
Give me a hint
Build first and second difference rows and test n(n + 1).
Compare my reasoning
- First differences: 4, 6, 8, 10.
- Second differences: 2, 2, 2, so the sequence is quadratic.
- n(n + 1) gives each listed value for n = 1 to 5.
First differences 4, 6, 8, 10; second differences 2; term n² + n.
Look for these in your work
- Difference rows accurate
- Quadratic pattern identified
- Rule verified
Rows of a theatre have 12, 15 and 18 seats in the first three rows. Model row n, predict row 20, and give one limitation.
Give me a hint
Fit the linear pattern, then consider physical constraints.
Compare my reasoning
- The common difference is 3, so begin with 3n.
- Add 9 to make the first term 12: seats = 3n + 9.
- Row 20 has 69 seats; a real theatre's width and aisles limit the model.
3n + 9 seats; row 20 has 69; venue design limits the pattern.
Look for these in your work
- Linear rule fitted
- Prediction correct
- Limitation is contextual
Common mistakes
- Confusing the next term with the nth term
- Using the wrong starting n
- Assuming quadratic from first differences
- Using differences for a constant-ratio sequence
What can you explain now?
Find the next term of 7, 11, 15, 19, …
Compare with the explanation
23
The constant difference is 4.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
What do constant first, second and third differences suggest, and how do you verify an nth-term rule?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.