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Fibonacci Numbers Recall that the Fibonacci numbers f(n) can be be defined recursively by letting f(0) = 0, f(1) = 1 and f(n) = f(n – 1) + f(n – 2) for all n > 2 € Zt. For all n > 2 E Z+, let S(n) be the statement: %3D V5 f(n) V5 Use strong induction to prove that S(n) holds for every positive integer n > 2. Hints: there may need to be multiple base cases. Also note: 3+y5 = (+y5)2 and -V5 3-V5 2

Fibonacci Numbers Recall that the Fibonacci numbers f(n) can be be defined recursively by letting f(0) = 0, f(1) = 1 and f(n) = f(n – 1) + f(n – 2) for all n > 2 € Zt. For all n > 2 E Z+, let S(n) be the statement: %3D V5 f(n) V5 Use strong induction to prove that S(n) holds for every positive integer n > 2. Hints: there may need to be multiple base cases. Also note: 3+y5 = (+y5)2 and -V5 3-V5 2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Fibonacci Numbers Recall that the Fibonacci numbers f(n) can be be defined recursively by letting f(0) = 0, f(1) =1 and f(n) = f(n – 1) + f(n – 2) for all n > 2 € Z+. For all n > 2 € Z+, let S(n) be the statement: %3D | | f(n) = V5 Use strong induction to prove that S(n) holds for every positive integer n > 2. Hints: there may need to be multiple base cases. Also note: 3+5 = (ty5)2 and (1월6)2 and -V5 3-V5 2
Transcribed Image Text:1. Fibonacci Numbers Recall that the Fibonacci numbers f(n) can be be defined recursively by letting f(0) = 0, f(1) =1 and f(n) = f(n – 1) + f(n – 2) for all n > 2 € Z+. For all n > 2 € Z+, let S(n) be the statement: %3D | | f(n) = V5 Use strong induction to prove that S(n) holds for every positive integer n > 2. Hints: there may need to be multiple base cases. Also note: 3+5 = (ty5)2 and (1월6)2 and -V5 3-V5 2
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