The fibonacci sequence is defined as F(n) = F(n-1) + F(n-2)for n>=2, F(0)=0, and

Question

The fibonacci sequence is defined as F(n) = F(n-1) + F(n-2)for n>=2, F(0)=0, and F(1)=1. Prove by inductionthe correct statement between thefollowing: 1. for n>=1, F(n)<=100(3/2)n 2. for n>=1, F(n)>=0.01(3/2)n Kindly give detailed steps for clearunderstanding. The fibonacci sequence is defined as F(n) = F(n-1) + F(n-2)for n>=2, F(0)=0, and F(1)=1. Prove by inductionthe correct statement between thefollowing: 1. for n>=1, F(n)<=100(3/2)n 2. for n>=1, F(n)>=0.01(3/2)n Kindly give detailed steps for clearunderstanding. Kindly give detailed steps for clearunderstanding.

Explanation / Answer

Dear.., Basis:
F(0)= 0, F(1)=1. F(2)=F(2-1)+F(2-2)        =F(1)+F(0)        =1 F(3) =F(2)+F(1)         =2 F(4) =F(3)+F(2)          =2+1         =3 here we clearly know that for n>=1,f(n)>=0.01(3/2)n istrue.
Induction: n=m its true when n>=2.. Suppose n=m-1 then F(m-1)+F(m-2) >= 0.01(3/2)m-1 F(m)=F(m-1)+F(m-2) >= 0.01(3/2)m-1                                  >=0.01(3)m-1/(2)m-1                                    >=0.01[(3)m-12-m+1]         Using the induction for n>=1, F(n)>=0.01(3/2)n is true.                                    >=0.01[(3)m-12-m+1]         Using the induction for n>=1, F(n)>=0.01(3/2)n is true.

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