Sequences (Copy)
Sequences Cheat Sheet (IGCSE Mathematics 0580 – CORE)
Topic: Number Sequences and nth Terms
1. Continue a Given Number Sequence or Pattern
- Look at how the numbers are changing (difference or rule).
- Add or subtract the same number = linear sequence.
- Increasing differences = non-linear (quadratic or cubic).
Examples:
- 2, 4, 6, 8, … → add 2 each time → next: 10, 12
- 1, 3, 6, 10, 15, … → add 2, 3, 4, 5… → next: 21, 28
2. Recognise Patterns in Sequences
a) Term-to-Term Rule:
- Rule to get from one term to the next
e.g. “Add 4” → 3, 7, 11, 15, …
b) Relationship Between Sequences:
- Compare sequences side-by-side or by difference
e.g.
Sequence A: 1, 4, 9, 16, 25 → square numbers (n²)
Sequence B: 2, 5, 10, 17 → difference increases by 2 each time → quadratic
3. Find and Use the nth Term of a Sequence
a) Linear Sequences (Constant Difference)
- nth term = an + b
where a = common difference, b = the term zero point
Steps to Find:
- Work out the difference between terms = a
- Use the 1st term to solve for b
Example: 4, 7, 10, 13, …
→ Difference = +3 → nth term = 3n + ?
→ When n = 1, term = 4 → 3(1) + b = 4 → b = 1
nth term = 3n + 1
b) Simple Quadratic Sequences (Increasing Difference)
- Form: nth term = an² + bn + c
Steps:
- Work out 1st differences, then 2nd differences
- 2nd difference ÷ 2 = a
- Use simultaneous equations or pattern matching to find b and c
Example: 2, 5, 10, 17, 26
- 1st differences = +3, +5, +7, +9
- 2nd differences = +2 → a = 1
Try nth term: n² + ?n + ?
Test:
1² = 1 → need 2 → +1
2² = 4 → need 5 → +1
3² = 9 → need 10 → +1
nth term = n² + 1
c) Simple Cubic Sequences (Rare at CORE, but may be recognised)
- Look at 3rd differences
- Example: 1, 8, 27, 64, … → Cube numbers: n³
Common Sequence Types and Their nth Terms
| Sequence Type | Example | nth Term Formula |
|---|---|---|
| Linear | 2, 4, 6, 8,… | 2n |
| Linear | 5, 8, 11, 14,… | 3n + 2 |
| Square Numbers | 1, 4, 9, 16,… | n² |
| Quadratic | 2, 5, 10, 17,… | n² + 1 |
| Triangular Numbers | 1, 3, 6, 10, 15,… | n(n + 1)/2 |
| Cube Numbers | 1, 8, 27, 64,… | n³ |
Key Tips
- Linear → same difference → nth term = an + b
- Quadratic → 2nd differences constant
- Always test your nth term formula with first 2 or 3 terms
- Don’t confuse term number (n) with the term value
CORE Exam Focus:
- Complete a sequence
- Find nth term (mostly linear, some quadratic)
- Identify term-to-term rule
- Recognise square, triangular, and cube number sequences
- Rarely: Use nth term to find a specific term (e.g. 10th term)