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.
Knowledge Organiser: Quadratic Sequences
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
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.