Key Terms
- Quadratic sequence: A sequence where the second differences are constant
- First differences: Differences between consecutive terms
- Second differences: Differences between the first differences
- nth term: Formula an² + bn + c giving any term from its position n
- Coefficient a: Half the second difference value
Must-Know Facts
- In a quadratic sequence the first differences change but the second differences are constant
- a = ½ × (second difference) — NOT the full second difference
- Subtract an² from each term to find the linear part bn + c
- Find b and c using the linear nth term method on the remainders
- Always check: substitute n = 1 and n = 2 to verify your formula gives the correct terms
- Square numbers: 1, 4, 9, 16, 25… (nth term = n²)
Key Formulas
- nth term = an² + bn + c
- a = (second difference) ÷ 2
- Subtract an² from each term → linear sequence → find bn + c
- n² sequence: 1, 4, 9, 16, 25 (second difference = 2, so a = 1)
Common Mistakes
- Using first difference instead of second: Must find second differences to identify a quadratic sequence
- a = second difference: Wrong — a = second difference ÷ 2
- Forgetting to subtract an²: After finding a, must subtract an² from each term before finding the linear part
- Sign errors: When second difference is negative, a is negative — check by substituting n=1,2,3
- Confusing quadratic with linear: If first differences are constant it is linear; only quadratic if second differences are constant
This topic summary covers Knowledge Organiser: Quadratic Sequences within Quadratic Sequences for GCSE Mathematics. Revise Quadratic Sequences in Algebra for GCSE Mathematics with 12 exam-style questions and 22 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 8 of 8 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Quadratic Sequences
Which of the following is a property of a quadratic sequence?
A student says: 'The sequence 3, 7, 13, 21, 31 is quadratic because the first differences increase.' Explain whether the student is correct and how to check properly.
Quick recall flashcards
What is a quadratic sequence?
A sequence where the second differences between consecutive terms are constant.
Example: 2, 5, 10, 17, 26, ...
First differences: 3, 5, 7, 9, ...
Second differences: 2, 2, 2, ... (constant)
What are second differences?
The differences between the first differences.
First differences: 4, 6, 8, 10
Second differences: 2, 2, 2
If second differences are constant, the sequence is quadratic.