1. The Fibonacci Numbers are the numbers defined by F0 = 0, F1 = 1, and Fn =
Fn−1+Fn−2 for n 2. (So for instance F2 = F1+F0 = 1+0 = 1, F3 = F2+F1 = 1+1 = 2,
and F4 = F3 + F2 = 2 + 1 = 3, etc.)
(a) Suppose that we set ~wn = (Fn, Fn−1) for n 1, and A =
[1 1]
[1 0] .
Show that the recursion relation above means that w(n+1)(subscript) = Aw(n)(subscript).
(b) Use the eigenvectors of A to find a formula for A^k(w1) for any k less than or equal to 0.
(c) Use the answer from (b) to find a formula for the n-th Fibonacci number Fn.

Question

1. The Fibonacci Numbers are the numbers defined by F0 = 0, F1 = 1, and Fn =
Fn−1+Fn−2 for n  2. (So for instance F2 = F1+F0 = 1+0 = 1, F3 = F2+F1 = 1+1 = 2,
and F4 = F3 + F2 = 2 + 1 = 3, etc.)
(a) Suppose that we set ~wn = (Fn, Fn−1) for n  1, and A = 
[1 1]
[1 0] .
Show that the recursion relation above means that w(n+1)(subscript) = Aw(n)(subscript).
(b) Use the eigenvectors of A to find a formula for A^k(w1) for any k less than or equal to 0.
(c) Use the answer from (b) to find a formula for the n-th Fibonacci number Fn.