The Fibonacci numbers form the sequence of numbers that begins 0, 1, 1, 2, 3, 5, 8, 13. If we let F_n denote the n-th Fibonacci number, then
F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, ...
In general, a Fibonacci number is the sum of the previous two Fibonacci numbers; that is, F_{n+2} = F_{n+1} + F_n. So we have
F_{n+2} = F_n + F_{n+1}.
a. If we write x_n = [F_{n+1}
F_n], find the matrix A such that x_{n+1} = A x_n.
b. Show that A has eigenvalues
lambda_1 = (1 + sqrt{5})/2 approx 1.61803...
lambda_2 = (1 - sqrt{5})/2 approx -0.61803...
with associated eigenvectors v_1 = [lambda_1
1] and v_2 = [lambda_2
1].
c. Classify this dynamical system as one of the six types seen in this section. What happens to x_n as n becomes very large?
d. Write the initial vector x_0 = [1
0] as a linear combination of eigenvectors v_1 and v_2.
e. Write the vector x_n as a linear combination of v_1 and v_2.
f. Explain why the n-th Fibonacci number
F_n = (1/sqrt{5})left(left(frac{1+sqrt{5}}{2}
ight)^n - left(frac{1-sqrt{5}}{2}
ight)^n
ight).
g. Use this relationship to compute F_{20}.
h. Explain why F_{n+1}/F_n approx lambda_1 when n is very large.
The number phi = (1+sqrt{5})/2 approx 1.61803 is called the golden ratio and is one of mathematics' special numbers.