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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

Πώς να το αναγνωρίζετε

• 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

Συνηθισμένα λάθη

• 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

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