Maths
What is a sequence in maths?
Quick answer
A sequence is a list of numbers or shapes that follows a rule, like 3, 7, 11, 15 (add 4 each time). Each item is called a term.
In one sentence
A sequence is a list of numbers or shapes in a set order that follows a rule, like 3, 7, 11, 15 (add 4 each time), so you can predict what comes next.
Explained
Look at 2, 4, 6, 8, 10. Each number goes up by 2, so the next one is 12. Each number in a sequence is called a term: 2 is the 1st term, 4 is the 2nd, and so on.
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Finding the rule
To continue a sequence, find the term-to-term rule: what gets you from one term to the next. The main types are:
- Add or subtract the same amount (an arithmetic or linear sequence): 7, 12, 17, 22 (add 5); 30, 26, 22, 18 (subtract 4). The amount is the common difference.
- Multiply or divide by the same number (a geometric sequence): 3, 6, 12, 24 (double); 1000, 100, 10, 1 (divide by 10).
- Special sequences: square numbers (1, 4, 9, 16…), triangular numbers (1, 3, 6, 10…, adding 2, then 3, then 4) and the Fibonacci sequence (1, 1, 2, 3, 5, 8…, where each term is the sum of the two before it). Fibonacci numbers turn up in nature, such as the spirals on many pinecones and sunflower heads.
Counting in steps is the same skill as your times tables: 4, 8, 12, 16 is the 4 times table, and 1, 3, 5, 7 are the odd numbers.
The nth term
In middle and high school (secondary school in the UK) you find a rule that jumps straight to any term. This is the position-to-term rule, or nth term, where n is the position.
Worked example: 5, 8, 11, 14…
- The difference is 3, so the rule starts with 3n.
- When n = 1, 3n is 3, but the real 1st term is 5. That is 2 more.
- So the nth term is 3n + 2.
- Check with the 4th term: 3 × 4 + 2 = 14. ✓
Now the 100th term is easy: 3 × 100 + 2 = 302. This is where sequences meet algebra.
Examples
| Sequence | Rule | Next term |
|---|---|---|
| 1, 3, 5, 7 | add 2 | 9 |
| 50, 45, 40, 35 | subtract 5 | 30 |
| 2, 6, 18, 54 | multiply by 3 | 162 |
| 1, 4, 9, 16 | square numbers | 25 |
| 1, 3, 6, 10 | add 2, 3, 4… | 15 |
Common mistakes
- Checking only the first jump. This is the most common slip. In 1, 2, 4, 8 it looks like "add 1", but the rule is "double". Test the rule on every pair.
- Mixing up the two kinds of rule. "Add 3" (term-to-term) is not the same as 3n + 2 (nth term).
Quick quiz
Sequences
Check what you know with 5 quick questions.
Question 1 of 5
What is the next term in 5, 8, 11, 14…?
Practise it
Try it
- What are the next two terms of 4, 9, 14, 19…?
- What is the rule for 81, 27, 9, 3…?
- Find the nth term of 6, 10, 14, 18…
Answers: 1) 24 and 29 (add 5). 2) Divide by 3, so the next term is 1. 3) 4n + 2 (check: 4 × 1 + 2 = 6).
Number pattern and sequence practice
Practise it step by step in a full course: Grade 6 Maths
For grown-ups
Related topics
Odd and even numbers
Even numbers can be split into two equal groups or pairs with none left over (2, 4, 6…). Odd numbers always leave one left over (1, 3, 5…).
Algebra
Algebra is maths that uses letters, like x or n, to stand for numbers we don't know yet or numbers that can change. For example, if n + 3 = 10, then n = 7.
Square numbers
A square number is what you get when you multiply a whole number by itself, like 4 × 4 = 16. The first square numbers are 1, 4, 9, 16, 25 and 36.
Times tables
Times tables are lists of the multiplication facts for a number, such as 3 × 1 = 3 up to 3 × 12 = 36. Knowing them by heart speeds up almost all later maths.
Compound interest and simple interest
Simple interest is paid only on the starting amount: I = P × r × t. Compound interest is also paid on earlier interest: A = P(1 + r)ⁿ. $1,000 at 10% for 2 years: $1,200 simple, $1,210 compound.
Equations
An equation says two things are equal, using an = sign, like 2x + 1 = 9. Solving it means finding the value of the letter that makes it true, here x = 4.