Deducing the nth term for linear and quadratic sequences [Higher tier]
The nth term lets you compute any term of a sequence directly. For arithmetic (linear) sequences this is straightforward; for quadratic sequences it requires a bit more algebra.
Linear (arithmetic) nth term
For a, a + d, a + 2d, …:
u_n = a + (n - 1)d = dn + (a - d).
So coefficient of n equals the common difference d; the constant is "first term minus d".
Example: 5, 8, 11, 14, 17, … d = 3, a = 5. u_n = 3n + (5 - 3) = 3n + 2.
Check: u_1 = 5 ✓, u_4 = 14 ✓.
Quadratic nth term
For a quadratic sequence with constant second difference Δ²:
- Coefficient of n² =
Δ²/2. - Subtract
(Δ²/2)n²from each original term to get a linear (arithmetic) residual. - Find the linear nth term of the residual, add it.
✦Worked example— Worked example — quadratic
Sequence: 4, 9, 18, 31, 48, … First differences: 5, 9, 13, 17. Second differences: 4 (constant).
So coefficient of n² = 4/2 = 2. Subtract 2n² from each original term:
- n=1: 4 - 2 = 2
- n=2: 9 - 8 = 1
- n=3: 18 - 18 = 0
- n=4: 31 - 32 = -1
- n=5: 48 - 50 = -2
Residual is 2, 1, 0, -1, -2 — arithmetic with d = -1, a = 2 → -n + 3.
Final nth term: u_n = 2n² - n + 3. Check: u_1 = 2 - 1 + 3 = 4 ✓; u_5 = 50 - 5 + 3 = 48 ✓.
Shortcut method (3-formula technique)
For quadratic with values u_1, u_2, u_3:
u_n = an² + bn + c.- Equations:
a + b + c = u_1;4a + 2b + c = u_2;9a + 3b + c = u_3. - Solve simultaneously.
This works but is slower than the differences method for typical GCSE.
Common patterns to recognise
n²→ 1, 4, 9, 16, 25n² + 1→ 2, 5, 10, 17, 26n² - n→ 0, 2, 6, 12, 20 — note this is 2 × triangular numbersn(n + 1)/2→ 1, 3, 6, 10, 15 — triangular
⚠Common mistakes— Common mistakes (examiner traps)
- Forgetting to halve the second difference for the n² coefficient.
- Solving the residual as quadratic. The point of the method is that the residual is linear — handle it with arithmetic nth term.
- Off-by-one indexing.
u_n = 3n + 2gives u_1 = 5; if you accidentally useu_0you'd get 2 (wrong start). - Missing the constant term. The constant comes from the residual's nth term, not from the original sequence's first term directly.
- Plugging into the wrong simultaneous equation. Double-check which row corresponds to which n.
➜Try this— Quick check
Sequence 3, 8, 15, 24, 35.
First diffs: 5, 7, 9, 11. Second diffs: 2 (constant). Coefficient of n² = 1.
Residual: 3 - 1, 8 - 4, 15 - 9, 24 - 16, 35 - 25 = 2, 4, 6, 8, 10 → 2n.
nth term: n² + 2n. Check: u_1 = 3, u_5 = 25 + 10 = 35 ✓.
AI-generated · claude-opus-4-7 · v3-deep-algebra