Fibonacci numbersThis problem develops properties of the Fibonacci numbers, which are defi...

Fibonacci numbers

This problem develops properties of the Fibonacci numbers, which are defined by recurrence (3.21). We shall use the technique of generating functions to solve the Fibonacci recurrence. Define the generating function (or formal power series) ℱ as

where Fi is the i th Fibonacci number.

a. Show that ℱ(z) = z + zℱ(z) + z2ℱ (z).

b. Show that

where

and

c. Show that

d. Prove that  rounded to the nearest integer. (Hint: Observe that )

e. Prove that