Miranda Coulter Math 2700 Spring 2010

From Fibonacci’s Liber 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 8 th month? The 24 th ? The n th ?

Time, n = Pairs, f n =

n fn fn n fn fn n fn fn n fn fn n fn fn Let f n denote the number of pairs at the beginning of month n. From this chart, we can see that f n = f n-1 + f n-2 when n > 2. For example, f 8 = f 7 + f 6 = = 21 But what expression gives f n for any n?

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

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

The next step is to find the eigenvalues of A.

We can now find the eigenvectors for the two eigenvalues.

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

Now plug it all in to (finally) obtain A n.

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