Q1 Step by step handwritten solution 1. Fibonacci NumbersRecall that the Fibonacci numbers f(n) can be be defined recursively by lettingf(0) = 0, f(1) =1 and f(n) = f(n – 1) + f(n – 2) for all n > 2 € Z+. For alln > 2 € Z+, let S(n) be the statement:%3D||f(n) =V5Use 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-V53-V52

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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 2 E Z+. For all n > 2 € Z+, let S(n) be the statement: %3D %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: 3ty5 = (1ty5)2 and 3-V5 %3D 2

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 2 E Z+. For all n > 2 € Z+, let S(n) be the statement: %3D %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: 3ty5 = (1ty5)2 and 3-V5 %3D 2

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.3: Divisibility

Problem 49E: 49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove...

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30. Prove statement of Theorem : for all integers .
49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove that if is a prime integer, then is a factor of for . (From Exercise of Section, it is known that is an integer.) b. Use induction on to prove that if is a prime integer, then is a factor of .

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