Arithmetic Sequences
An arithmetic sequence has a constant difference between consecutive terms, so its graph is a straight line of points.
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An arithmetic sequence has a constant difference between consecutive terms, so its graph is a straight line of points.
- The nth term is a + (n-1)d
- 3, 7, 11, 15 has difference 4
- The sum is n over 2 times (first + last)
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Arithmetic Sequences?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Arithmetic 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 Arithmetic Sequences in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Arithmetic Sequences gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Arithmetic Sequences in detail
An arithmetic sequence goes up or down by the same amount each time, called the common difference d. For 5, 8, 11, 14 the difference is 3. The nth term is a + (n − 1)d, where a is the first term, which simplifies here to 3n + 2.
To find the rule, work out the common difference, which becomes the coefficient of n, then adjust with a constant so the first term matches. You can test whether a number belongs to the sequence by solving for n. Arithmetic sequences model steady savings, constant speed and evenly spaced seating.
How the interactive helps you understand Arithmetic Sequences
Move the slope and y-intercept sliders. The line and its equation update, so you can see how the slope sets the steepness and the intercept sets where the line crosses the y-axis.
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 Arithmetic Sequences.
Real-life uses and examples
- Spot Arithmetic Sequences in exam questions, where it often sits inside a short word problem.
- Check a homework answer about Arithmetic Sequences by estimating first, then working it out step by step.
- Use Arithmetic Sequences when comparing prices, discounts or recipe quantities in everyday life.
Common mistakes to avoid
- Forgetting units, signs or brackets when you apply Arithmetic Sequences.
- Memorising the steps for Arithmetic Sequences without checking why they work.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about Arithmetic Sequences
What is Arithmetic Sequences?
An arithmetic sequence has a constant difference between consecutive terms, so its graph is a straight line of points.
What are the most important points about Arithmetic Sequences?
The main points are: The nth term is a + (n-1)d; 3, 7, 11, 15 has difference 4; The sum is n over 2 times (first + last).
Why is Arithmetic Sequences useful?
Understanding Arithmetic Sequences gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Arithmetic 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 Arithmetic Sequences?
Related topics in Mathematics include Algebraic Expressions, Factorising, Rearranging Formulae.