Quadratic sequences
Objective. Recognise constant second differences.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A quadratic sequence is a sequence whose terms follow a rule involving n², so the differences between consecutive terms are not constant but the second differences are. To identify a quadratic sequence, calculate the first differences between terms, then the differences of those differences; a constant second difference confirms the sequence is quadratic. For example, the sequence 2, 5, 10, 17, 26 has first differences 3, 5, 7, 9 and second differences all equal to 2, confirming it is quadratic with an n² coefficient of 2 ÷ 2 = 1. This halved second difference gives the coefficient of n² in the nth term formula.
Worked example
Find the second difference of the quadratic sequence 4, 9, 16, 25, 36 to confirm it is quadratic.
- 1Find the first differences: 9−4=5, 16−9=7, 25−16=9, 36−25=11.
- 2The first differences are 5, 7, 9, 11, which are not constant.
- 3Find the second differences: 7−5=2, 9−7=2, 11−9=2.
- 4All second differences equal 2, confirming the sequence is quadratic.
Answer: The second difference is constant and equal to 2
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Pupils sometimes stop after finding constant first differences and wrongly call the sequence linear, without checking whether those first differences are actually still changing. Always compute second differences to test for a quadratic pattern.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.