Fibonacci Numbers and Generating Functions - psholtz/MIT-SICP GitHub Wiki
This problem develops properties of the Fibonacci numbers, which are defined by recurrence. We shall use the technique of generating functions to solve the Fibonacci recurrence. Define the generating function (or formal power series) F as:
a. Show that F(z) = z + zF(z) + z^2F(z).
b. Show that
where
and
c. Show that
d. Prove that F(i) = phi^i/sqrt(5), rounded to the nearest integer.
e. Prove that F(i+2) >= phi^i for i >= 0.
Solution
We first demonstrate that F(z) = z + zF(z) + z^2F(z) as follows.