Geometric Sequences
A geometric sequence has a constant ratio between consecutive terms, so it grows or shrinks exponentially.
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A geometric sequence has a constant ratio between consecutive terms, so it grows or shrinks exponentially.
- Each term is the previous times r
- 2, 6, 18, 54 has ratio 3
- The nth term is a times r to the n-1
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Geometric Sequences?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Geometric Sequences. Maths builds step by step, so each topic here gives you a rule, a picture and a quick check you can use on homework and in exams.
Check that you can explain Geometric Sequences in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Geometric Sequences gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Geometric Sequences in detail
A geometric sequence multiplies by the same number each time, called the common ratio r. The sequence 3, 6, 12, 24 has r = 2 and the sequence 81, 27, 9, 3 has r = 1/3. The nth term is arⁿ⁻¹, where a is the first term.
Geometric sequences model situations with percentage growth or decay, such as compound interest, population growth, bouncing balls and the spread of a virus. They grow or shrink far faster than arithmetic sequences, which is the source of exponential behaviour that you will meet in graphs later.
How the interactive helps you understand Geometric Sequences
Watch the growth curve in the interactive. Use any sliders or buttons it offers to change the values and see how the growth curve responds.
Before you change anything, predict the result. Then change one value at a time and check whether your prediction was right. This is a quick way to build a correct mental model of Geometric Sequences.
Real-life uses and examples
- Explain Geometric Sequences to a friend with a simple picture, such as a row of tiles or a bag of sweets.
- Spot Geometric Sequences in exam questions, where it often sits inside a short word problem.
- Check a homework answer about Geometric Sequences by estimating first, then working it out step by step.
Common mistakes to avoid
- Memorising the steps for Geometric Sequences without checking why they work.
- Forgetting units, signs or brackets when you apply Geometric Sequences.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about Geometric Sequences
What is Geometric Sequences?
A geometric sequence has a constant ratio between consecutive terms, so it grows or shrinks exponentially.
What are the most important points about Geometric Sequences?
The main points are: Each term is the previous times r; 2, 6, 18, 54 has ratio 3; The nth term is a times r to the n-1.
Why is Geometric Sequences useful?
Understanding Geometric Sequences gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Geometric Sequences?
Open the interactive at the top of this page and change the controls one at a time. Predict the result before each change, then take the three-question quiz. A score of two or more marks the topic as complete on this device.
Which topics are related to Geometric Sequences?
Related topics in Mathematics include Substitution, Linear Equations, Simultaneous Equations.