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Solving Fibonacci

This piece explores a classic problem from Fibonacci's "Liber Abaci," detailing how a pair of rabbits reproduces over time. We analyze the growth of the rabbit population over several months, demonstrating the Fibonacci sequence's fundamental concept: each number is the sum of the two preceding ones. By examining this rabbit population problem, we can compute the number of pairs at any given month, including the eighth and twenty-fourth months, using matrix diagonalization techniques. This method provides a comprehensive approach to solving Fibonacci-related problems.

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Solving Fibonacci

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  1. Solving Fibonacci Miranda Coulter Math 2700 Spring 2010

  2. From Fibonacci’sLiber Abaci, Chapter 12 How Many Pairs of Rabbits Are Created by One Pair in One Year A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. How many rabbit pairs would there be in the 8th month? The 24th? The nth? History of the Fibonacci Sequence

  3. Time, n = 1 2 3 4 5 6 1 1 Pairs, fn = 2 3 5 8

  4. Let fn denote the number of pairs at the beginning of month n. From this chart, we can see that fn= fn-1+ fn-2 when n > 2. For example, f8 = f7+ f6 = 13 + 8 = 21 But what expression gives fn for any n?

  5. By multiplying this matrix repeatedly, the nth and (n+1)th term can be found.

  6. To simplify An, a technique known as Diagonalization must be used. With this method An can be written using the matrices P, P-1, and D (a diagonal matrix).

  7. The next step is to find the eigenvaluesof A.

  8. We can now find the eigenvectors for the two eigenvalues.

  9. With values for λ, we can now construct P, P-1, and D.

  10. Now plug it all in to (finally) obtain An.

  11. After hours of confusing and grueling matrix multiplication, the formula for the nth term of the Fibonacci sequence finally emerges.

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