Second Differences

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A pattern where the differences between consecutive terms form their own arithmetic sequence.

Second differences occur when the first differences between terms aren't constant, but the differences of those differences (second differences) are constant. This indicates a quadratic relationship. Sequences with constant second differences include square numbers, triangular numbers, and other polynomial sequences of degree 2.

If second differences are constant = c, then aₙ = An² + Bn + C

• 1, 4, 9, 16, 25... (squares) → 1st diff: 3, 5, 7, 9 → 2nd diff: 2, 2, 2
• 1, 3, 6, 10, 15... (triangular) → 1st diff: 2, 3, 4, 5 → 2nd diff: 1, 1, 1
• 2, 6, 12, 20, 30... → 1st diff: 4, 6, 8, 10 → 2nd diff: 2, 2, 2

How to recognize it

• Calculate first differences between consecutive terms
• If first differences aren't constant, calculate second differences
• Constant second differences indicate a quadratic pattern
• The sequence grows faster than arithmetic but slower than geometric

Common mistakes

• Stopping at first differences when they're not constant
• Confusing with geometric sequences (check ratios vs differences)
• Forgetting to continue to third differences for cubic sequences

Practice guidance

Number Sequences