GCSE MATHS • HIGHER TIER
Geometric Sequences Made Simple
Recognise repeated multiplication, find common ratios, form nth term rules and calculate specific terms.
Clear method
Exam practice
Worked proofs
Geometric Sequences in Four Moves
1
Identify
Look at how each term changes from one term to the next.
2
Find the Ratio
Divide one term by the previous term and check that the ratio stays constant.
3
Form the Rule
Use the first term and common ratio in the geometric nth term formula.
4
Substitute
Use the nth term rule to find any required term.
What You’ll Learn
Understand what a geometric sequence is.
Find and use the common ratio.
Compare arithmetic and geometric sequences.
Continue growing and decaying geometric sequences.
Find nth term rules for geometric sequences.
Use nth term rules to find specific terms.
Key Rules
Geometric sequence
Each term is found by multiplying the previous term by the same number.
Common ratio
Common Ratio = Next Term ÷ Previous Term
nth term formula
nth Term = First Term × (Common Ratio)⁽ⁿ⁻¹⁾
Growth or decay
If the ratio is greater than 1, the sequence grows. A ratio between 0 and 1 gives geometric decay.
Worked Proof: Geometric Sequences
Step-by-Step Method
- Check how the terms change from one term to the next.
- Divide each term by the previous term and check that the answer stays the same.
- Use First Term × (Common Ratio)⁽ⁿ⁻¹⁾.
- Replace n with the required term number and calculate the answer.
More Worked Examples
1
Finding the Next Term
Multiply the last term by the common ratio.
Find the next term: 4, 12, 36, 108
Common Ratio = 3
Common Ratio = 3
2
The nth Term of a Geometric Sequence
The nth term formula is: First Term × (Common Ratio)^(n−1)
Sequence: 2, 6, 18, 54
First Term = 2
Common Ratio = 3
First Term = 2
Common Ratio = 3
3
Finding Specific Terms
Substitute the term number into the nth term formula.
nth Term = 2 × 3^(n−1)
Find the 5th term.
Find the 5th term.
4
Geometric Decay
100, 50, 25, 12.5
Common Ratio = 0.5
Visual Growth vs Decay Diagram
Growth: 2 → 4 → 8 → 16
Decay: 100 → 50 → 25 → 12.5
Decay: 100 → 50 → 25 → 12.5
- Common GCSE Mistakes
- Confusing arithmetic and geometric sequences.
- Using subtraction instead of division when finding ratios.
- Using incorrect powers in nth term formulas.
- Making calculator mistakes.
- Forgetting that n−1 is used in the exponent.
- Exam Tips
- Check for a constant ratio before calling a sequence geometric.
- Use division, not subtraction, to find the common ratio.
- Remember the nth term uses the exponent n−1.
- Check whether the sequence is growing or decaying.
- Use powers carefully when substituting term numbers.
- Compare arithmetic and geometric sequences if you are unsure.
Skills Used in Geometric Sequence
Sequences
Common Ratio
nth Term
Exponential Growth
Geometric Decay
Substitution
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Printable Worksheet
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Interactive Quiz
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Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Geometric Sequence Tutorial
Step-by-step walkthroughs of Geometric Sequence with clear methods and exam tips.
Frequently Asked Questions
It is a sequence in which each term is found by multiplying the previous term by the same number.
It is the fixed multiplier used from one term to the next.
Divide consecutive terms. If the ratio stays constant, the sequence is geometric.
Arithmetic sequences use a common difference, while geometric sequences use a common ratio.
Yes. A common ratio such as 0.5 produces geometric decay.
First Term × (Common Ratio)^(n−1).