Zweite Differenzen

Lernen

Ein Muster, bei dem die Differenzen aufeinanderfolgender Glieder selbst eine arithmetische Folge bilden.

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

So erkennen Sie es

• 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

Häufige Fehler

• 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

Übungshinweise

Zahlenreihen