Bonjours à tous,
J'aurais besoin d'aide pour la dernière question de cet exercice (désolé l'énoncé est en anglais car il est tiré d'une épreuve de section europèenne) :
The earliest mathematical model of population growth can be found in the work of Leonardo
of Pisa, in 1220. [. . .] It was about the reproductive behaviour of rabbits. Not in its biological
sense, but numerological. Leonardo took as the basic unit a pair of rabbits - a natural enough
hypothesis. Assume that in the beginning there is one pair of immature rabbits. These mature for
a season. Every season after, they beget one immature pair, which in turn matures for a season.
And of course, all newly mature pairs beget1 one immature pair per season as well. Suppose that
rabbits and their procreative urges never die. How many pairs of rabbits will have been begotten
after n seasons ?
Suppose there are Mn mature pairs and In immature pairs in season n. Then we start out in
season 1 with M1 = 0, I1 = 1. The growth laws are :
In+1 = Mn and Mn+1 = Mn + In.
From Does God play dice ? by Ian Stewart
Questions
1. What is the difference between an immature pair and a mature one ?
2. Explain the growth laws given at the end of the text.
3. In a table, compute the values of Mn and In for n from 1 to 8.
4. Let Tn be the total number of pairs of rabbits in season n. Compute the values of Tn for n
from 1 to 8.
5. Prove that for any natural number n, Tn+2 = Tn+1 + Tn and deduce that (Tn+2/Tn+1)*(Tn+1/Tn)=Tn+1/Tn+1 +1
6. We admit that the ratio Tn+1/Tn
approaches a positive real number q when n approaches +1.
a. Explain why we can say that q is a solution of the equation x^2− x − 1 = 0.
b. Compute the exact value of q.
Voici ce que j'ai essayé de faire pour la question 6 :
En partant de Tn+1/Tn=(TTn+2/Tn)-1, J'ai remplacé en utilisant Tn=Mn+In puis en utilisant les relation de l'énoncé j'ai tenté d'aboutir à une expression ne comportant que des Mn ou des In mais je n'arrive à rien de concluant.
Merci de votre aide.
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