State clearly and explicitly where and how you are using Theorem 2. (a) The Fibonacci numbers are recursively defined by 0 = 1

State clearly and explicitly where and how you are using Theorem 2.

(a) The Fibonacci numbers are recursively defined by α₀ = α₁ = 1, α_n+1 = α_n + α_n-1 if n = 1, 2, · · ·. Find the limit of the sequence (α_n+1/α_n).
(b) Compute a list of α₁, · · ·, α₁₂. Show that α₁₂ = 233 is the number of pairs of rabbits after 12 months if initially there is 1 pair and each pair generates 1 pair per month, beginning in the second month of existence (no deaths occurring).
(c) Show that the generating function of the Fibonacci numbers is f(z) = 1/(1 - z - z²); that is, if a power series (1) represents this f(z), its coefficients must be the Fibonacci numbers and conversely. Start from f(z)(1 - 1 - z²) = 1 and use Theorem 2.