Quadratic nth terms

Objective. Derive nth-term rules from constant second differences.

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Learn: the key idea

A quadratic nth-term rule is a formula un = an² + bn + c that generates a sequence whose second differences are constant. Find the first differences, then the second differences; the second difference is always 2a, so a = (second difference)÷2. Subtract a·n² from the original sequence to leave a linear sequence, then find that linear rule as usual for bn + c. For example 2, 7, 14, 23, 34 has first differences 5, 7, 9, 11 and second difference 2, so a = 1; subtracting n² gives 1, 3, 5, 7, 9, which is 2n − 1, so un = n² + 2n − 1. Always check your formula against at least two terms of the original sequence.

Worked example

Find the nth-term rule for the quadratic sequence 4, 9, 16, 25, 36.

1 $First differences: 5, 7, 9, 11; second difference = 2, so a = 2\div2 = 1$
2 $Subtract n²: 4-1=3, 9-4=5, 16-9=7, 25-16=9, 36-25=11$
3 $Linear part 3, 5, 7, 9, 11 has nth term 2n + 1$
4 $Combine: un = n² + 2n + 1, check n=1: 1+2+1=4 ✓, n=2: 4+4+1=9 ✓$

Answer: un = n² + 2n + 1

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Common mistake. Pupils use the second difference itself as a, forgetting to halve it. For 2, 7, 14, 23, 34 the second difference is 2, so a = 1, not a = 2; using a = 2 gives wrong terms. Always divide the constant second difference by two first.

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