Quick revision notes · Mathematics 0580 · Chapter 15 of 37
Number Sequences
Term-to-term rules, nth terms of linear, special, quadratic, cubic and exponential sequences, and shape patterns.
6 sectionsWorked examples
Common mistakes
At a glance
Term-to-term rules, nth terms of linear, special, quadratic, cubic and exponential sequences, and shape patterns.
1Term-to-term rules
- The term-to-term rule says how to get from one term to the next: 3, 7, 11, 15 → add 4.
- Arithmetic (linear) sequences add the same each time; geometric ones multiply by the same each time.
- Look at the differences: constant first difference → linear; constant second difference → quadratic.
Worked example
Find the next two terms of 2, 6, 18, 54, …
- Each term × 3.
162, 486
2Linear sequences: the nth term
- nth term = dn + (first term − d), where d is the common difference.
- To test whether a number is a term, set the nth term equal to it: a whole-number n means yes.
- To find which term a value is, solve nth term = value.
Formulas
nth term = dn + (a − d)a = first term, d = difference
Worked example
Find the nth term of 7, 11, 15, 19, … Is 101 a term?
- d = 4: 4n. 4n gives 4, 8, 12…; add 3.
- nth term 4n + 3.
- 4n + 3 = 101 → n = 24.5, not a whole number.
4n + 3; 101 is not a term
3Special sequences
- Recognise these and their shifts (+ or −) and multiples:
| Sequence | Terms | nth term |
|---|---|---|
| Square | 1, 4, 9, 16, 25 | n2 |
| Cube | 1, 8, 27, 64, 125 | n3 |
| Triangle | 1, 3, 6, 10, 15 | n(n + 1)/2 |
| Powers of 2 | 2, 4, 8, 16, 32 | 2n |
| Square + 1 | 2, 5, 10, 17, 26 | n2 + 1 |
| Twice square | 2, 8, 18, 32, 50 | 2n2 |
Worked example
Find the nth term of 0, 3, 8, 15, 24, …
- Compare with squares 1, 4, 9, 16, 25: each is 1 less.
n2 − 1
4Quadratic sequences
- If the second difference is constant, the nth term is an2 + bn + c.
- a = second difference ÷ 2.
How to do it
- Find the first and second differences.
- a = second difference ÷ 2.
- Subtract an2 from each term.
- Find the nth term of what is left (linear) and add it on.
Worked example
Find the nth term of 4, 10, 18, 28, 40, …
- Differences 6, 8, 10, 12; second difference 2, so a = 1.
- Subtract n2 (1, 4, 9, 16, 25): 3, 6, 9, 12, 15 = 3n.
n2 + 3n
5Cubic and exponential sequences
- Constant third difference → cubic; often a simple form like n3 + 1.
- A constant ratio → exponential: nth term = a × rn−1 (a = first term, r = ratio).
- Combinations: compare with the basic sequence and spot the change.
Formulas
nth term = a × rn−1geometric (exponential)
Worked example
Find the nth term of 5, 10, 20, 40, …
- Ratio r = 2, first term 5.
5 × 2n−1
6Patterns of shapes
- Count the items in the first few patterns and write them as a sequence.
- Find the nth term, then use it to predict any pattern or to find which pattern uses a given number.
Worked example
Pattern n uses 3n + 1 matchsticks. Which pattern uses 100 matchsticks?
- 3n + 1 = 100.
- 3n = 99.
Pattern 33
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| nth term of 5, 8, 11 is n + 3 | The difference is 3, so 3n + 2 |
| Using the second difference as a | a is HALF the second difference |
| Saying 101 is the 24.5th term | n must be a whole number, so 101 is not a term |
| 5 × 2n for 5, 10, 20, 40 | Check n = 1: 5 × 2n−1 gives 5 |
★In the exam
- Always check your nth term with n = 1 and n = 2.
- Show the differences — they earn method marks.
- Give the nth term in simplest form (3n + 2, not 3n + 5 − 3).