Prove Theorem 9.8: For each positive integer n, the nth Fibonacci number is Fn = n1 0 _
Chapter 9, Problem 18(choose chapter or problem)
Prove Theorem 9.8: For each positive integer n, the nth Fibonacci number is
\(F_{n}=\left\{\begin{array}{ll} \left(\begin{array}{c} n-1 \\ 0 \end{array}\right)+\left(\begin{array}{c} n-2 \\ 1 \end{array}\right)+\left(\begin{array}{c} n-3 \\ 2 \end{array}\right)+\cdots+\left(\begin{array}{c} k \\ k-1 \end{array}\right) & \text { if } n=2 k \\ \left(\begin{array}{c} n-1 \\ 0 \end{array}\right)+\left(\begin{array}{c} n-2 \\ 1 \end{array}\right)+\left(\begin{array}{c} n-3 \\ 2 \end{array}\right)+\cdots+\left(\begin{array}{c} k \\ k \end{array}\right) & \text { if } n=2 k+1 . \end{array}\right.\)
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