Question 1. Let (F,) be a sequence defined by Fy = Fo = 1, F,, = F,,_1 + F,,2 where n > 2. Prove by induction that for any n N, n > 1, i+ F+...+F,=F,;021. Question 2. Use induction to show that for any natural n > 3, one can be written as a sum of n 1 distinct fractions of the form where k is a natural number. 1 1 1 E le. F =3 1l=-+-+=. DKOITTIE: ERb 1 1 5173 ': 6 Hint. Note that = pd + _b(b D This is useful for the induction step. Question 3. Consider a sequence ay,as,as,... with a; = 1, ao = 3 and a;, = a;_1 + a2 for all keN, k>3 7\\ " an 1. Prove by induction that if n is divisible by 3, then F}, is even