Sequences and the nth term
A sequence is a list of numbers that follows a rule. A term-to-term rule gets you the next number; an nth term rule gets you any term straight away, such as the 100th, without writing out the 99 before it.
Part 1 of 3: Learn it
In short
- Linear sequence: nth term = (common difference) × n + (zero term).
- To test whether a number is in a sequence, set the nth term equal to it and see if n is a whole number.
- Quadratic sequence: the second differences are constant, and half of that gives the n² coefficient.
Where this is in your specification
Spec points: DfE A23, A24 and A25 (the nth term of a quadratic sequence is Higher tier)
| Board | Topic: Sequences |
|---|---|
| DfE content | A23, A24, A25 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | 6 Algebra |
| Eduqas C300 | HA23 to HA25 |
| Cambridge IGCSE 0580 | E2.7 |
Kinds of sequence
| Sequence | Example | Rule |
|---|---|---|
| Arithmetic (linear) | 7, 11, 15, 19 | add the same amount each time |
| Geometric | 3, 6, 12, 24 | multiply by the same amount each time |
| Fibonacci-type | 1, 1, 2, 3, 5, 8 | add the two previous terms |
| Square numbers | 1, 4, 9, 16 | n² |
| Triangular numbers | 1, 3, 6, 10 | add 2, then 3, then 4 |
The nth term of a linear sequence
- Find the common difference. For 7, 11, 15, 19 it is 4, so the rule starts 4n.
- Compare with the 4 times table (4, 8, 12, 16): each term is 3 more.
- So the nth term is 4n + 3. Check: n = 1 gives 7.
A quicker way to the number on the end: it is the term before the first one, called the zero term. 7 − 4 = 3.
Is a number in the sequence?
Set the nth term equal to the number and solve. If n comes out as a whole number, it is a term. For 4n + 3 = 101, 4n = 98 and n = 24.5, so 101 is not in the sequence.
Quadratic sequences (Higher)
If the first differences change but the second differences are all the same, the nth term has an n² term. Its coefficient is half the second difference. Subtract that n² part from each term, and what is left is a linear sequence: find its nth term and add it on.
Find the nth term of 5, 12, 19, 26.
Show the answer
Difference 7, zero term −2, so 7n − 2.
Part 2 of 3: See it worked
Worked examples
Example 1
Find the nth term of 20, 17, 14, 11, ... and its 30th term.
- Common difference: −3, so −3n
- Zero term: 20 + 3 = 23
- nth term = 23 − 3n; 30th term = 23 − 90
Answer: 23 − 3n, and the 30th term is −67.
Example 2
Find the nth term of 3, 9, 19, 33, 51. (Higher)
- First differences: 6, 10, 14, 18. Second differences: 4, so the n² coefficient is 2
- 2n²: 2, 8, 18, 32, 50
- Sequence − 2n²: 1, 1, 1, 1, 1
Answer: 2n² + 1.
Common mistakes
- Writing n + 4 instead of 4n when the difference is 4.
- Using the first term as the number on the end: 7, 11, 15 is 4n + 3, not 4n + 7.
- Saying a number is in a sequence because the answer for n is close to a whole number.
- In a quadratic sequence, using the full second difference instead of half of it as the n² coefficient.
Find the 50th term of the sequence with nth term 6n − 1.
Show the answer
6 × 50 − 1 = 299.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Find the nth term of 5, 12, 19, 26.
Difference 7, zero term −2, so 7n − 2.
Find the 50th term of the sequence with nth term 6n − 1.
6 × 50 − 1 = 299.
Is 80 a term of the sequence 3n + 2?
3n + 2 = 80 gives n = 26, a whole number, so yes: it is the 26th term.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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