Fibonacci

# Fibonacci - Harvard CS 121 CSCI E-207 September 6 2011 An...

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September 6, 2011 An inductive proof Proposition: The n ’th Fibonacci number is the number of strings over { a,b } of length n - 2 with no consecutive b ’s. First step: make sure you know what everything means What are the Fibonacci numbers? Where does the indexing start? How does the statement of the proposition need to be cleaned up to be precise? Is the proposition true for the smallest value of n ? If it is true for small values of n , say for all n k , why would it be true for n = k + 1 ? 1

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September 6, 2011 So here is what we want to prove: Proposition: For every n 2 , F n is the number of strings over { a,b } of length n - 2 with no consecutive b ’s. Proof: Base case: Prove it for n = 2 and n = 3 , by a hand check. Induction step: For any n , let X n be the set of strings of length n with no consecutive b ’s. We have proven that | X n | = F n +2 for n = 0 and n = 1 . Now let k be any number 1 and assume we have proven that | X n | = F n +2 for all n k . We want to conclude that | X
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Fibonacci - Harvard CS 121 CSCI E-207 September 6 2011 An...

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