The Fibonacci numbers f1, f2, f3,... are defined by setting f1 = f2 = 1 and fn = fn-1 + fn-2 for all integers n ≥ 3. Prove by induction on n that gcd(fn, fn-1) = 1 for all integers n > 2. Prove that for all integers n ≥ 3, gcd(fn, fn-1) = gcd (fn-1, fn-2). n Prove by induction on n that Σf² = fn+1fn for all integers n ≥ 1. i=1

The Fibonacci numbers f1, f2, f3,... are defined by setting f1 = f2 = 1 and fn = fn-1 + fn-2 for all integers n ≥ 3. Prove by induction on n that gcd(fn, fn-1) = 1 for all integers n > 2. Prove that for all integers n ≥ 3, gcd(fn, fn-1) = gcd (fn-1, fn-2). n Prove by induction on n that Σf² = fn+1fn for all integers n ≥ 1. i=1

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 46E: Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection...

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