In a quadratic sequence, the differences change, but the second differences are constant. For 0, 3, 10, 21, 36, the first differences are 3, 7, 11, 15 and the second differences are all 4.
The coefficient of n² is half the second difference: here 2n². Subtract 2n² from each term (2, 8, 18, 32, 50) to get −2, −5, −8, −11, −14, which is linear: −3n + 1. So the nth term is 2n² − 3n + 1.
Worked example
Find the nth term of 2, 6, 12, 20, …
- Second difference 2, so start with n².
- Terms − n²: 1, 2, 3, 4, which is n.
Answer:n² + n
Key words
- second difference
- the difference between differences
- quadratic sequence
- nth term contains n²
- linear remainder
- what is left after subtracting the n² part
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