Exercise 4.4.5 ($\sum_{k=1}^n F_k = F_{n+1} - 1$)

Prove that F 1 + F 2 + F 3 + ⋯ + F n = F n + 2 − 1 .

Answers

Proof. We can reason by induction, but here we prefer to use the usual properties of sums. Starting from F k = F k + 2 − F k + 1 for all k ∈ ℕ , we obtain

∑ k = 1 n F k = ∑ k = 1 n ( F k + 2 − F k + 1 ) = ∑ k = 1 n F k + 2 − ∑ k = 1 n F k + 1 = ∑ j = 2 n + 1 F j + 1 − ∑ j = 1 n F j + 1 ( j = k + 1  in the first sum ) = F n + 2 + ∑ j = 2 n F j + 1 − ∑ j = 2 n F j + 1 − F 2 = F n + 2 − 1 .

For all positive integer n ,

∑ k = 1 n F k = F n + 2 − 1 .

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2025-02-06 08:36
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