3. The Fibonacci numbers, which we have discussed in class, are a sequence defined by the rules F₁ = 1, F2 = 1, and Fn = Fn-1Fn-2 for n ≥2. This problem is about the quantity Gn=Fn+2Fn+1-Fn+3Fn. (a) Compute Gn for a few values of n and make a guess about what it is in general. (b) Write down the following two expressions: • Gn+1, but with Fn+4 rewritten in terms of Fn+3 and Fn+2; Gn, but with Fn rewritten in terms of Fn+1 and Fn+2. ● 1 Now both expressions are written entirely in terms of Fn+1, Fn+2, and Fn+3, which helps see the connection between them. (c) What is the connection between the results you get, and how could you use it to prove the guess you made in part (a)?

3. The Fibonacci numbers, which we have discussed in class, are a sequence defined by the rules F₁ = 1, F2 = 1, and Fn = Fn-1Fn-2 for n ≥2. This problem is about the quantity Gn=Fn+2Fn+1-Fn+3Fn. (a) Compute Gn for a few values of n and make a guess about what it is in general. (b) Write down the following two expressions: • Gn+1, but with Fn+4 rewritten in terms of Fn+3 and Fn+2; Gn, but with Fn rewritten in terms of Fn+1 and Fn+2. ● 1 Now both expressions are written entirely in terms of Fn+1, Fn+2, and Fn+3, which helps see the connection between them. (c) What is the connection between the results you get, and how could you use it to prove the guess you made in part (a)?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 28E

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