Fibonacci's Rabbits Leonardo Fibonacci da Pisa (c. 1170-1250), an outstanding European mathematician of the middle ages stated an interesting problem in his main work Liber Abaci (1202) where he wanted to know how many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair brings forth a new pair that becomes productive from the second month on and assuming no pairs die. Therefore, starting with F1 = 1 newly born pair in the first month, we can find the number Fk of pairs in the kth month. Using some knowledge of calculus, acquired when studying sequences, we can determine the number of pairs of rabbits in the kth month by setting up the following equation: Fk = (Number of pairs alive the preceding month) + (Number of newly born pairs for the kth month) Since the rabbits do not produce offsprings during the first month of their lives, we can see that the number of newly born pairs for the kth month is the number Fk - 2 of pairs alive two months before. Thus we can write the equation above as Fk = Fk - 1 + Fk - 2 Fibonacci's relation If we set F0= 0, denoting 0 pairs for month 0 before the coming of the first newly born pairs of rabbit, then we obtain the sequence F0, F1, F2, ...,Fk, ... for the number of pairs of rabbits, which becomes the Fibonacci's sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …, where each term starting with F2 = 0 + 1 = 1 is the sum of the two preceding terms. Therefore, for any particular value of k, we could compute Fk by writing out the sequence far enough. Question: Using the above information, describe carefully your method of calculating the eigenvectors and eigenvalues and determine the number of pairs of rabbits for the 7th, 10th, and 13th months.
Fibonacci's Rabbits
Leonardo Fibonacci da Pisa (c. 1170-1250), an outstanding European mathematician of the middle ages stated an interesting problem in his main work Liber Abaci (1202) where he wanted to know how many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair brings forth a new pair that becomes productive from the second month on and assuming no pairs die. Therefore, starting with F1 = 1 newly born pair in the first month, we can find the number Fk of pairs in the kth month.
Using some knowledge of calculus, acquired when studying sequences, we can determine the number of pairs of rabbits in the kth month by setting up the following equation:
Fk = (Number of pairs alive the preceding month) + (Number of newly born pairs for the kth month)
Since the rabbits do not produce offsprings during the first month of their lives, we can see that the number of newly born pairs for the kth month is the number Fk - 2 of pairs alive two months before. Thus we can write the equation above as
|
Fk = Fk - 1 + Fk - 2 |
Fibonacci's relation |
If we set F0= 0, denoting 0 pairs for month 0 before the coming of the first newly born pairs of rabbit, then we obtain the sequence
F0, F1, F2, ...,Fk, ...
for the number of pairs of rabbits, which becomes the Fibonacci's sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …,
where each term starting with F2 = 0 + 1 = 1 is the sum of the two preceding terms.
Therefore, for any particular value of k, we could compute Fk by writing out the sequence far enough.
Question:
Using the above information, describe carefully your method of calculating the eigenvectors and eigenvalues and determine the number of pairs of rabbits for the 7th, 10th, and 13th months.
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