(1) Prove each of the following identities for $n \ge 1$ in two ways: one inductive and one combinatorial. (a) $\sum_{i=1}^{n} F_i = F_{n+2} - 1$. (b) $\sum_{i=1}^{n} F_{2i} = F_{2n+1} - 1$. (c) $\sum_{i=1}^{n} F_{2i-1} = F_{2n}$. (2) Prove that if $k, n \in \mathbb{P}$ with $k \mid n$ (meaning $k$ divides evenly into $n$) then $F_k \mid F_n$.
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Inductive hypothesis: Assume Fi = Fn+2 - 1 for some i > 1. Inductive step: We want to show that Fi+1 = Fn+3 - 1. Using the definition of Fibonacci sequence, we have Fi+1 = Fi + Fi-1. By the inductive hypothesis, Fi = Fn+2 - 1 and Fi-1 = Fn+1 - 1. Show more…
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