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Teilbarkeitsaussage
Guided induction proof of a divisibility statement (with moderate randomization).
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said | Ready to use | 4 years, 6 months ago |
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Ulrich Görtz 4 years, 6 months ago
Published this.Ulrich Görtz 4 years, 6 months ago
Gave some feedback: Ready to use
Ulrich Görtz 4 years, 6 months ago
Created this.There is only one version of this question that you have access to.
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a | integer |
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b | integer |
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r | integer |
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k | integer |
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Generated value: list
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Beweisen Sie durch vollständige Induktion, dass für alle natürlichen Zahlen n die Zahl {b}{{r}n+2{k}}−{la−1}n durch {a} teilbar ist.
Behauptung: Für alle natürlichen Zahlen n ist {b}{{r}n+{2rk}}−{la−1}n durch {a} teilbar.
Beweis durch vollständige Induktion nach n.
Induktionsanfang
Induktionsschritt n>0. Wir müssen nun zeigen, dass {b}{{r}n+{2rk}}−{la−1}n durch {a} teilbar ist, und dürfen dabei benutzen:
Wir schreiben den Ausdruck, um den es geht, so um, dass wir den Term aus der Induktionsvoraussetzung dort wiederfinden:
{b}{{r}n+{2rk}}−{la−1}n={b}{{r}(n−1)+{2rk}}⋅{b}{r}−{la−1}n
={b}{{r}(n−1)+{2rk}}⋅{b}{r}−{la−1}n−1⋅{b}{r}+
wobei A:=
{la−1}n−1⋅{b}{r}−{la−1}n={la−1}n−1⋅(
und dies ist offensichtlich ein Vielfaches von {a}, weil {{b}{r}}−{{la−1}}={{b}{r}−{la−1}}={{b}{r}−{la−1}{a}}⋅{a} ein Vielfaches von {a} ist.
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