2.0 Number Theory Combinatorics

Let \( F_k \) be the \( k \)-th Fibonacci number defined by \( F_1 = 1 \), \( F_2 = 1 \), and \( F_k = F_{k-1} + F_{k-2} \) for \( k \geq 3 \).
Prove by mathematical induction that for all positive integers \( n \):
$$ \sum_{i=1}^n F_i = F_{n+2} - 1 $$

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