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Samenvatting Inleveropdracht uitwerking week 12 fundamenten van de wiskunde
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Inleveropdracht uitwerking week 12 fundamenten van de wiskunde
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Fundamenten uitwerkingen inleveropgave week 1218 december 2021
Inleveropgave:
Gegeven en te doen opschrijven: 1pt
Gegeven: a ∈ R>0 .
Te doen: stelling 2.5.5 of stelling 2.5.11 gebruiken om f : N → R gegeven door f (n) = an! + n
recursief te definiëren.
Welke stelling passen we toe en hoe?: 1pt
We zullen stelling 2.5.11 toepassen met H = R en f zal de functie zijn die de plaats van
functie g inneemt.
Waarmee beginnen we?: 1pt
We merken op dat f (1) = a1! + 1 = a1 + 1 = a + 1, hiermee neemt a + 1 ∈ R de plaats
in van e in deze stelling.
Functie t definieren : 2pt
We definieren nu de functie t : R × N → R als f (x, y) = (x − y)y+1 + y + 1.
Laten zien dat t van de juiste vorm is : 2pt
Als f (n) = an! + n dan geldt
t(f (n), n) = (f (n) − n) · an+1 + n + 1 = (an! + n − n)n+1 + n + 1
= (an! )n+1 + n + 1 = an!·n+1 + n + 1 = a(n+1)! + n + 1 = f (n + 1)
Hiermee is de functie t van de juiste vorm.
Daadwerkelijk de conclusie van Stelling 2.5.11 gebruiken : 1pt
Stelling 2.5.11. geeft nu dat f de unieke functie is zodanig dat f (1) = a + 1 en
f (n + 1) = t(f (n), n).
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