State clearly and explicitly where and how you are using Theorem 2 (a) The Fibonacci numbers are recursively defined by 0 1 1, n 1 n n 1 if n 1, 2, Find the limit of the sequence ( n 1 n ) (b) Compute a list of 1 , , 12 Show that 12 233 is the number of pairs of rabbits after 12 months if initially...

a Division of the recursion relation by n gives Take the ...

State clearly and explicitly where and how you are using Theorem 2. (a) The Fibonacci numbers are

Question:

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-1if 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.