GCSE-readyQuadratic sequences and the nth term
In a quadratic sequence the differences change, but the differences of the differences stay the same.
Spot it
First differences: 7, 11, 15, 19
Second differences: 4, 4, 4
A constant second difference means the nth term is an2 + bn + c.
Find a
2n2 gives: 2, 8, 18, 32, 50
Find the linear part
This is linear: n + 1.
So the nth term is 2n2 + n + 1. Check n = 3: 22.
Find the nth term of 4, 11, 22, 37, 56, ...
- Second difference 4, so a = 2.
- Sequence − 2n2: 2, 3, 4, 5, 6
- That is n + 1.
Answer: 2n2 + n + 1
Second difference ÷ 2 gives the n2 coefficient. Subtract that n2 part, and find the nth term of the linear sequence left over.
- Quadratic
- An expression or equation where the highest power of x is 2, such as x^{2} − 5x + 6.
- nth term
- A rule that gives any term of a sequence from its position n, such as 3n + 2.
- Term
- A single number, letter or product of them in an expression, such as 5x or −3.
- Coefficient
- The number in front of a letter in an algebraic term. In 7x^{2}, the coefficient is 7.
All GCSE and International GCSE Mathematics key terms and definitions →
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2 more questions
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Find the 20th term of the sequence with nth term n2 + 6n.
Show the answer
Answer: 520
400 + 120 = 520.
Find the second difference of 6, 21, 42, 69, ...
Show the answer
Answer: 6
First differences 15, 21, 27; second difference 6.
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Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.