Question

(5) Separate N into two disjoint sets: the evens E, and the odds O. Consider the set of Fibonacci ). Prove (n F and En F are infinite sets,6 numbers {1, 1, 2, 3, 5, 8, 13x13 21x21 8x8 Figure 1.10: An interesting geometric proof could use a patterns of the Fibonacci spiral, although there are simpler proofs. the

Answer 1

We will prove this using a formula for Fibonacci series which is: Proof of the formula: We will only prove the inductive step. Consider that it is true Vn < k. Now consider n-k +1. Then, Fb Back to the question: Observe that from the formula we can see that if Fn-3 is odd then so is Fn+3. Similarly it holds if Fn-3 is even. Then observe that we can create two infinite sequences {F3nJnEN and F3n+ilnEN the first one being odd and 2nd one being even. Now observe that {F3n),EN En F and {F3n +1}rIEN є F as F4-3 є O and F3-2 є E. Hence as we found two infinite divergent sequence, one in On F, one in En F, therefore we can say that both On F and En F are infinite.