Build arithmetic and geometric sequence rules
Use a constant difference or ratio to predict terms · Write simple nth-term rules · Explain why a short pattern is not a uniquely determined sequence
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Negative-number arithmetic
- Integer powers
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Use a constant difference or ratio to predict terms
- Write simple nth-term rules
- Explain why a short pattern is not a uniquely determined sequence
Arithmetic sequences
An arithmetic sequence adds the same difference d at each step. Number terms from n = 1: uₙ = a + (n − 1)d, where a is the first term. Check the first term and the common difference against the data.
The nth term names one term; it is not a sum. A sequence can decrease when d is negative.
Geometric sequences
A geometric sequence multiplies by the same ratio r. With first term a, uₙ = arⁿ⁻¹. Negative ratios alternate signs. A ratio between zero and one gives decreasing positive terms when a > 0.
A few listed terms do not prove an infinite rule uniquely. State that the arithmetic or geometric pattern continues; check every supplied transition before using the model.
Sources for this lesson
Pause and explain
For 5, 8, 11, … numbered from 1, what is the nth term?
Worked example
Assuming the patterns continue, find the sixth terms of 4, 9, 14, … and 3, 6, 12, ….
Show the worked solution
- The first difference is 5: u₆ = 4 + 5×5 = 29.
- The second ratio is 2: v₆ = 3×2⁵ = 96.
Answer 29 and 96, respectively.
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.
Assuming an arithmetic pattern, find uₙ and u₁₀ for 7, 11, 15, ….
Give me a hint
Find d and substitute n = 1 as a check.
Compare my reasoning
- d = 4, so uₙ = 7 + 4(n − 1) = 4n + 3.
- u₁₀ = 43.
uₙ = 4n + 3; u₁₀ = 43.
Look for these in your work
- Check the first term
- Distinguish n from n − 1
Assuming a geometric pattern, find vₙ and v₅ for 8, −4, 2, ….
Give me a hint
Divide consecutive non-zero terms to find r.
Compare my reasoning
- r = −1/2, so vₙ = 8(−1/2)ⁿ⁻¹.
- v₅ = 8/16 = 1/2.
vₙ = 8(−1/2)ⁿ⁻¹; v₅ = 1/2.
Look for these in your work
- Retain the negative ratio
- Check sign and exponent
A plan gives 12 points on day 1 and 3 more than the previous day thereafter. How many on day 8, and on which day are 39 awarded?
Give me a hint
Write one formula for the stated arithmetic rule.
Compare my reasoning
- uₙ = 12 + 3(n − 1) = 3n + 9.
- u₈ = 33. Solve 3n + 9 = 39 to get n = 10.
33 on day 8; 39 on day 10.
Look for these in your work
- Interpret a term as one day's award
- Require a positive integer day
Common mistakes
- The first exponent is n − 1, not n.
- A nth-term formula needs an indexing convention and a pattern assumption.
What can you explain now?
For 5, 8, 11, … numbered from 1, what is the nth term?
Compare with the explanation
3n + 2.
5 + (n − 1)3 simplifies to 3n + 2.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
For 5, 8, 11, … numbered from 1, what is the nth term?
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.