Algebra · GCSE Maths
Quadratic sequences
GCSE Maths quadratic sequences: constant second difference, nth term an² + bn + c — find a from half the second difference, then b and c from the first two terms.
Linear sequences have a constant first difference. Quadratic sequences have a constant second difference. Half that second difference is a in an² + bn + c.
The important bits
What you need to know
- 1
A quadratic sequence has a constant second difference. 2, 6, 12, 20, 30 has first differences 4, 6, 8, 10 and second difference 2.
- 2
The nth term has the form an² + bn + c. For a quadratic sequence, a = second difference ÷ 2.
- 3
Method: write the sequence. Find first differences. Find second differences. a = half the constant second difference.
- 4
Build a table of an² using your a. Subtract an² from each term to leave a linear sequence bn + c. Find its nth term as usual.
- 5
Alternatively, use simultaneous equations with n = 1, 2, 3: substitute three terms into an² + bn + c and solve.
- 6
To test whether 200 is in the sequence 2n² + 1, solve 2n² + 1 = 200. n² = 99.5, so n is not an integer — 200 is not a term.
- 7
If n is a positive integer, the number is in the sequence. n starts at 1 for the first term unless the question says otherwise.
- 8
Do not use the linear “dn + (first − d)” rule on a quadratic; the first difference is not constant, so that rule finds the wrong nth term.
Quotations worth analysing
Short evidence. Real method.
“nth term = an² + bn + c”
a comes from the second difference. b and c come from the leftover linear part or from three simultaneous equations. This is not an + b.
“a = second difference ÷ 2”
Second difference 6 means a = 3, so the n² column is 3n². Forgetting to halve produces 6n² and every later line is wrong.
“Subtract an² to get a linear sequence”
Once an² is removed, bn + c is a linear nth term problem. Students who stop at an² + 3n without finding c miss the constant term.
Go deeper
Second difference tells you the n² coefficient
Sequence 3, 8, 15, 24, 35. First differences: 5, 7, 9, 11. Second differences: 2, 2, 2. a = 2 ÷ 2 = 1, so the n² part is n². Compute n²: 1, 4, 9, 16, 25. Subtract from the sequence: 3 − 1 = 2, 8 − 4 = 4, 15 − 9 = 6, 24 − 16 = 8, 35 − 25 = 10. The leftovers 2, 4, 6, 8, 10 are linear with d = 2, nth term 2n. Full nth term: n² + 2n. Check n = 1: 1 + 2 = 3. n = 4: 16 + 8 = 24. If the second difference is −4, a = −2 and the parabola opens downward in the difference table. The method is identical; the sign of a follows the second difference.
Go deeper
Simultaneous equations when the table feels heavy
For terms 5, 8, 13 at n = 1, 2, 3: a + b + c = 5, 4a + 2b + c = 8, 9a + 3b + c = 13. Subtract equations to eliminate c. (4a + 2b + c) − (a + b + c) = 8 − 5 gives 3a + b = 3. (9a + 3b + c) − (4a + 2b + c) = 13 − 8 gives 5a + b = 5. Subtract again: 2a = 2, a = 1. Then b = 0, c = 4, nth term n² + 4. Three equations, three unknowns — mechanical but safe when the second-difference route is muddled. Always check with n = 2 and n = 3 after you find a, b and c.
Go deeper
Is a number in the sequence?
Set an² + bn + c equal to the target and solve for n. You need a positive integer. 2n² + 3 = 50 gives 2n² = 47, n² = 23.5, not a square — not in the sequence. n² + 2n = 48 gives n² + 2n − 48 = 0, (n + 8)(n − 6) = 0, n = 6 or n = −8. Only n = 6 is a valid position, so 48 is the 6th term. Questions that ask “which term” want the value of n, not the term value. Questions that ask “is 100 in the sequence?” want yes/no with the algebra showing whether n is an integer. Do not confuse quadratic nth term with quadratic formula unless you rearranged to = 0 on purpose.
See the idea in action
Find the nth term of 2, 6, 12, 20, 30, … Step 1: First differences: 4, 6, 8, 10. Second differences: 2, 2, 2. Step 2: a = 2 ÷ 2 = 1, so n² part is n². Step 3: n² values: 1, 4, 9, 16, 25. Subtract: 2 − 1 = 1, 6 − 4 = 2, 12 − 9 = 3, 20 − 16 = 4, 30 − 25 = 5. Step 4: Leftovers 1, 2, 3, 4, 5 are linear with nth term n. Step 5: Full nth term: n² + n. Check: n = 3 gives 9 + 3 = 12. n = 5 gives 25 + 5 = 30.
Exam technique
Turn knowledge into marks
Write the second difference and circle it. Halve it for a before you hunt b and c. A wrong a poisons every line after.
Common mistakes
Do not give these marks away
- 01
Using the linear nth-term rule on a sequence whose second difference is constant.
- 02
Forgetting to halve the second difference, so a is double what it should be.
- 03
Stopping at an² + bn without finding c, or subtracting an² incorrectly in the reduction table.
A quadratic sequence has constant second difference 8. The coefficient a in an² + bn + c is
A8
B4
C16
D2
Show the answer
4. a = second difference ÷ 2 = 4. 8 is the second difference itself. 16 would be double. 2 would be a quarter.
Quick questions
If this is the bit you searched
How do I know if a sequence is quadratic?
The first differences are not constant, but the second differences are. If second differences are not constant, the sequence is not quadratic on the GCSE model.
Can the nth term be just n²?
Yes, if b = 0 and c = 0. The sequence 1, 4, 9, 16 has nth term n². Most exam sequences also have a linear or constant piece.
Is this Higher only?
Quadratic nth terms are Higher tier on most GCSE papers. Foundation has linear sequences. Check your specification.
What if the sequence starts at n = 0?
GCSE nth terms usually use n = 1 for the first term. If the question defines n differently, follow its definition and adjust the table.