These are proofs please answer all questions and parts and show the work neatly. 1. Use generalized induction to prove that n! < n^2 for all integers n ≥ 2. 2. The Fibonacci sequence{fn}= 1,1,2,3,5,8,13,21, . . .is defined recursively byfn+1=fn+fn−1for alln≥2, wheref1=f2= 1. In each case, use complete induction (witha= 1,b= 2) to prove the given statement.(a) Prove thatfn<2nfor all integersn≥1. (b) Prove thatfn=(1 +√5)n−(1−√5)n2n√5for all integersn≥1.

These are proofs please answer all questions and parts and show the work neatly. 1. Use generalized induction to prove that n! < n^2 for all integers n ≥ 2. 2. The Fibonacci sequence{fn}= 1,1,2,3,5,8,13,21, . . .is defined recursively byfn+1=fn+fn−1for alln≥2, wheref1=f2= 1. In each case, use complete induction (witha= 1,b= 2) to prove the given statement.(a) Prove thatfn<2nfor all integersn≥1. (b) Prove thatfn=(1 +√5)n−(1−√5)n2n√5for all integersn≥1.

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter8: Sequences And Series

Section8.5: Mathematical Induction

Problem 33E

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Question

These are proofs please answer all questions and parts and show the work neatly.

1. Use generalized induction to prove that n! < n^2 for all integers n ≥ 2.

2. The Fibonacci sequence{fn}= 1,1,2,3,5,8,13,21, . . .is defined recursively byfn+1=fn+fn−1for alln≥2, wheref1=f2= 1. In each case, use complete induction (witha= 1,b= 2) to prove the given statement.(a) Prove thatfn<2nfor all integersn≥1. (b) Prove thatfn=(1 +√5)n−(1−√5)n2n√5for all integersn≥1.

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