Definitiontwoquantitiesaandbaresaidtohavethegoldenrati

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Unformatted text preview: The Fibonacci Sequence and Pythagoras’ Theorem While it may not be obvious that there is any connection between the Fibonacci Sequence and Pythagoras’ Theorem, the following result shows that we can generate infinitely many triples ‐ though not all of them, for example (7, 24, 25) ‐ using Fibonacci numbers. Theorem: If Fn represents the nth Fibonacci number, then for n ≥ 1 (Fn Fn+3)2 + (2Fn+1 Fn+2)2 = (2Fn+1 Fn+2 + Fn2)2 Proof: The Golden Ratio Before stating and proving the well known connection with the Fibonacci sequence, we need to define the Golden Ratio (sometimes denoted by φ) and explain its historical significance. Definition: Two quantities a and b are said to have the golden ratio if Euclid gave this definition in t...
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This note was uploaded on 09/22/2011 for the course MAC 2311 taught by Professor Evinson during the Spring '08 term at University of Central Florida.

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