Quadratic sequences

For uₙ = an² + bn + c, the constant second difference is 2a. Use this to find a, subtract an² from the sequence, then fit a linear rule to the remainder. State whether indexing begins at n = 0 or n = 1.

A few terms do not uniquely determine an arbitrary sequence. The quadratic assumption is part of the task. Once a rule is found, substitute the given indices to check all supplied terms.

Worked example

Find a quadratic rule for 3, 8, 15, 24, …, with first term n = 1.

1. First differences are 5, 7, 9; second difference is 2, so a = 1.
2. Subtract n²: the remainder is 2, 4, 6, 8.
3. The remainder has rule 2n.

Answer: uₙ = n² + 2n

Revise first: Expanding and factorising.