Never Mix Up Arithmetic and Geometric Sequences Again – A Clear Guide for Exam Success

When working through past paper questions, it’s common for students to get confused between arithmetic and geometric sequences. Both types of sequences appear frequently in A Level exams, and knowing how to distinguish between them is crucial for picking the right formulas and methods. This blog post and study guide will help you confidently identify and work with each type, so you can avoid common mistakes and maximise your marks.

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Understanding the Differences

Let’s start by summarising the key features of arithmetic and geometric sequences in a clear table:

Feature	Arithmetic Sequence	Geometric Sequence
Definition	Each term is found by adding a constant	Each term is found by multiplying by a
	difference to the previous term	constant ratio to the previous term
Common Name	Linear sequence	Exponential sequence
Formula for nth term	a_n = a_1 + (n-1)d	a_n = a_1 \times r^{n-1}
Key Parameter	Common difference (d)	Common ratio (r)
Example	2, 5, 8, 11, 14, … (add 3 each time)	3, 6, 12, 24, 48, … (multiply by 2 each time)
Sum of first n terms	S_n = \frac{n}{2}[2a_1 + (n-1)d]	S_n = a_1 \frac{1 - r^n}{1 - r} (if r \neq 1)

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How to Spot the Difference in Exam Questions

1. Check the Pattern:
   • If the difference between consecutive terms is always the same, it’s arithmetic.
   • If the ratio (divide one term by the previous) is always the same, it’s geometric.
2. Look for Keywords:
   • Words like “constant difference” or “increases by” suggest arithmetic.
   • Words like “multiplied by” or “constant ratio” suggest geometric.
3. Test with the First Few Terms:
   • Subtract consecutive terms: if the result is constant, it’s arithmetic.
   • Divide consecutive terms: if the result is constant, it’s geometric.

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Common Mistakes to Avoid

• Mixing up formulas: Always double-check which sequence you’re dealing with before using a formula.
• Assuming all sequences are arithmetic: Not all sequences with a pattern are arithmetic—always test for a common ratio as well.
• Forgetting the first term: Both formulas require the first term (), so make sure you identify it correctly.

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Quick Practice

Given the sequence: 4, 8, 16, 32, …

• Subtract: 8-4 = 4, 16-8 = 8 (not constant)
• Divide: 8/4 = 2, 16/8 = 2 (constant ratio)
  Conclusion: This is a geometric sequence.

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Final Tip

When revising, try to create your own examples and test yourself. If you need more structured support, an A Level Maths Revision Course can help you master these concepts with expert guidance and plenty of practice.

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Summary Table for Quick Reference

Sequence Type	How to Identify	nth Term Formula	Example
Arithmetic	Constant difference	a_n = a_1 + (n-1)d	2, 5, 8, 11…
Geometric	Constant ratio	a_n = a_1 + (n-1)d	3, 6, 12, 24…

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With these tips and the summary table, you’ll be able to confidently tackle any sequence question in your exam. Good luck with your studies!