Recognising sequence families
The exam expects you to identify a sequence by sight and know its standard properties. Memorise the first few terms of each family — that recognition saves vast amounts of time.
Arithmetic sequences
Common difference d (constant). nth term: u_n = a + (n - 1)d, where a is the first term.
Examples: 4, 7, 10, 13, … (a = 4, d = 3); 20, 18, 16, 14, … (a = 20, d = -2).
Geometric sequences
Common ratio r (constant). nth term: u_n = a × r^{n-1}.
Examples: 2, 6, 18, 54, … (r = 3); 64, 32, 16, 8, … (r = ½).
To check: divide consecutive terms and see if the ratio is constant.
Triangular numbers
T_n = n(n + 1)/2. First few: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.
Visual: rows of dots forming triangles. Differences: 2, 3, 4, 5, … (increase by 1 each step).
Square numbers
S_n = n². First few: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Cube numbers
C_n = n³. First few: 1, 8, 27, 64, 125, 216.
Fibonacci sequence
u_1 = 1, u_2 = 1, u_{n+1} = u_n + u_{n-1}. First few: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.
Variants exist with different starting pairs — same recurrence.
Quadratic sequences
The second differences are constant (and non-zero). First differences are linear; second differences pin down the n² coefficient as half of that. Example: 4, 7, 12, 19, 28. Differences: 3, 5, 7, 9. Second differences: 2 (constant). Quadratic.
How to identify quickly
- Compute first differences.
- Constant → arithmetic (linear nth term).
- Linear (changing by a fixed amount) → quadratic.
- Multiplicative (each term × constant) → geometric.
- Check standard families:
- 1, 4, 9, 16, 25 → squares.
- 1, 8, 27, 64 → cubes.
- 1, 3, 6, 10, 15 → triangular.
- 1, 1, 2, 3, 5, 8 → Fibonacci.
✦Worked example— Worked example — classify
3, 9, 27, 81, 243 — common ratio 3 → geometric.
1, 4, 9, 16, 25 — perfect squares → squares (special quadratic).
5, 8, 11, 14 — constant difference 3 → arithmetic.
1, 4, 10, 19, 31 — first differences 3, 6, 9, 12 (linear); second differences 3 (constant) → quadratic.
⚠Common mistakes— Common mistakes (examiner traps)
- Confusing arithmetic with quadratic. Constant first differences = arithmetic; constant second differences = quadratic.
- Misidentifying geometric. A geometric sequence has constant ratio, not constant difference.
- Forgetting that 1 is the first term of squares, cubes, triangular, Fibonacci. Don't double-count by starting at 0.
- Mixing up Fibonacci variants. The classic begins 1, 1; some textbooks use 0, 1.
- Treating cubes as geometric. Cubes go 1, 8, 27 — ratios 8, 3.375 (not constant). They're a power sequence, not geometric.
➜Try this— Quick check
6, 9, 14, 21, 30 — first differences 3, 5, 7, 9 (linear); second differences 2 → quadratic.
AI-generated · claude-opus-4-7 · v3-deep-algebra