Recognise arithmetic and geometric progressions
Identify whether a sequence has a constant difference or constant ratio, and distinguish an AP from a GP.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What stays constant in an arithmetic progression?
What you will learn
- Recognise arithmetic and geometric progressions
- Find a common difference or common ratio
- Explain why a sequence is neither type when its pattern is not constant
Arithmetic progressions add the same difference
An arithmetic progression (AP) has a constant difference d between consecutive terms. Subtract successive terms to check: 5, 9, 13, 17 has d = 4.
A negative common difference is allowed. The sequence 23, 19, 15, 11 has d = −4, so it is still an AP even though it decreases.
Geometric progressions multiply by the same ratio
A geometric progression (GP) has a constant ratio r between consecutive terms, found by dividing each term by the previous one, provided the previous term is nonzero. The sequence 3, 6, 12, 24 has r = 2.
A sequence is neither type if its differences are not all equal and its ratios are not all equal. Check the defining pattern rather than deciding from whether the terms increase or decrease.
Pause and explain
What is the common ratio of 5, 10, 20, 40?
Worked example
Classify 4, 7, 10, 13 and 2, 6, 18, 54.
Show the worked solution
- For the first sequence, consecutive differences are 3, 3 and 3.
- For the second, consecutive ratios are 3, 3 and 3.
- A constant difference identifies an AP; a constant ratio identifies a GP.
Answer 4, 7, 10, 13 is an AP with d = 3; 2, 6, 18, 54 is a GP with r = 3.
Try a topic test
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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.
Classify 7, 11, 15, 19, and state its common difference.
Give me a hint
Subtract each term from the next one.
Compare my reasoning
- 11 − 7 = 4.
- 15 − 11 = 4 and 19 − 15 = 4.
- The difference is constant, so this is an AP.
AP with common difference 4.
Look for these in your work
- Successive differences calculated
- Pattern checked throughout
- Classification justified
Classify 81, 27, 9, 3, and state its common ratio.
Give me a hint
Divide each term by the term before it.
Compare my reasoning
- 27/81 = 1/3.
- 9/27 = 1/3 and 3/9 = 1/3.
- The ratio is constant, so this is a GP.
GP with common ratio 1/3.
Look for these in your work
- Ratios calculated
- Constant multiplier verified
- Classification justified
A sequence is 2, 5, 10, 17, 26. Is it an AP, a GP, or neither? Show the check.
Give me a hint
Compare both successive differences and, if useful, successive ratios.
Compare my reasoning
- The differences are 3, 5, 7 and 9, so they are not constant.
- The ratios are also not constant.
- Therefore it is neither an AP nor a GP.
Neither: the consecutive differences and ratios are not constant.
Look for these in your work
- Differences checked
- Ratios not assumed constant
- Conclusion supported
Common mistakes
- Checking only the first difference or ratio
- Calling every increasing sequence a GP
- Dividing by zero to test a ratio
- Assuming a negative difference prevents a sequence from being an AP
What can you explain now?
Is 12, 8, 4, 0 an AP or GP?
Compare with the explanation
AP with d = −4
Each consecutive difference is −4.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How can subtraction and division distinguish an AP from a GP?
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.