Arithmetic Sequences
Each term differs from the last by a constant.
About this topic
An arithmetic sequence is a list of numbers in which each term differs from the previous one by a constant called the common difference d. Examples are 3, 7, 11, 15, … with d = 4. The nth term is aₙ = a₁ + (n − 1)d.
Arithmetic sequences are linear functions whose inputs are whole numbers, so the graph is a set of dots lying on a straight line with slope d. To find the 20th term of 3, 7, 11, …, use a₂₀ = 3 + 19 · 4 = 79. You can also work backward: if the 5th term is 17 and the 8th is 26, then 3d = 9, so d = 3 and the first term is 5.
Many real patterns are arithmetic: seats in rows of a theater that increase by a fixed number, a savings plan adding the same deposit each month, or stacked logs in a pile. Recognizing the pattern lets you predict any later term without listing them all.
Common mistakes include using n instead of n − 1 in the formula and mixing up the difference with the ratio. Check by subtracting consecutive terms; if the answer is always the same, the sequence is arithmetic.
Key ideas
- aₙ = a₁ + (n − 1)d
- d is the common difference
- Graph is a set of collinear points
Quick check
Use cases: where Arithmetic Sequences helps
- Homework: review the key idea behind Arithmetic Sequences before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Arithmetic Sequences without notes.
- Visual learning: play with the interactive lab to see how Arithmetic Sequences changes when you adjust its inputs.
- Teaching a friend: explain Arithmetic Sequences in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Arithmetic Sequences?
Each term differs from the last by a constant.
What will I learn in Arithmetic Sequences?
aₙ = a₁ + (n − 1)d. d is the common difference. Graph is a set of collinear points.
Is Arithmetic Sequences part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Arithmetic Sequences lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Arithmetic Sequences?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.