Question: Let ( Fn) be a sequence defined by F1 = F2 = 1, Fn = Fn-1 + Fn-2 where n > 2. Prove by induction

Let ( Fn) be a sequence defined by F1 = F2 = 1, Fn = Fn-1 + Fn-2 where n > 2. Prove by induction that for any n E N, n > 1, F1 + F2 + ... + Fn = Fn+2- 1.Use induction to show that for any natural n 2 3, one can be written as a sum of n distinct fractions of the form - where k is a k natural number. Example. For n = 3, 1 = + + 6 Hint. Note that + This is useful for the induction step. b +1 b(b + 1)

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