State clearly and explicitly where and how you are using Theorem 2. (a) The Fibonacci numbers are
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State clearly and explicitly where and how you are using Theorem 2.
(a) The Fibonacci numbers are recursively defined by α0 = α1 = 1, αn+1 = αn + αn-1 if n = 1, 2, · · ·. Find the limit of the sequence (αn+1/αn).
(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.
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