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I would like some help in this problem:

given the fibonacci sequence 1,2,3,5,8,13....

we know that a_n/a_(n-1) --> phi when n --> [tex]\infty[/tex]

I have tried a new proof in these terms:

My goal is to prove that when n --> [tex]\infty[/tex], a_n/a_(n-1) = a_(n+1)/a_n in the limit

I have notice that a_n^2= a_(n-1)*a_(n+1) [tex]\pm[/tex] 1 for some values

Dividing both sides of the equation by a_n*a_(n-1)

[tex]\frac{a_n}{a_n_-_1}[/tex] = [tex]\frac{a_n_+_1}{a_n}[/tex][tex]\pm[/tex] [tex]\frac{1}{a_n*a_n_-_1}[/tex]

I want to find the limits, because the second term in the right side of the equation --> 0 when n --> [tex]\infty[/tex]

But I don't know if the equation above is valid for all n, so I'll prove this by induction as follows:

supose that the expression above is valid for some n

by definition a_(n+1)=a_n + a_(n-1)

a_n^2= a_(n-1)*[a_n + a_(n-1)] [tex]\pm[/tex] 1 ==> a_n^2 - a_(n-1)^2= a_(n-1)*a_n [tex]\pm[/tex] 1 ==>

==> [a_n + a_(n-1)]*[a_n - a_(n-1)] = a_(n-1)*a_n [tex]\pm[/tex] 1

by definition a_n - a_(n-1)= a_(n-2) ==>

==> [a_n - a_(n-1)]* a_(n-2) = a_(n-1)*a_n [tex]\pm[/tex] 1

by definition a_n = a_(n-1) + a_(n-2) ==>

==> [a_n - a_(n-1)]* a_(n-2) = a_(n-1)*[a_(n-1) + a_(n-2)] [tex]\pm[/tex] 1 ==>

==> a_(n-2)*a_n + a_(n-1)*a_(n-2) = a_(n-1)^2 + a_(n-1)*a_(n-2) [tex]\pm[/tex] 1 ==>

==> a_(n-1)^2 = a_(n-2)*a_n [tex]\pm[/tex] 1

compare the two expressions

a_n^2= a_(n-1)*a_(n+1) [tex]\pm[/tex] 1

a_(n-1)^2 = a_(n-2)*a_n [tex]\pm[/tex] 1

most generally we have

[tex]\{a}{_i}{^2}[/tex]} = [tex]\{a_i_-_1*a_i_+_1}[/tex]} [tex]\pm[/tex] 1, with i=2,3,4,5,6...,n

proving by induction (sorry about the english and notation)

Hence,

Lim[tex]\{_n_-_>_i_n_f_i_n_i_t_y}[/tex][tex]\frac{a_n}{a_n_-_1}[/tex]} = Lim[tex]\{_n_-_>_i_n_f_i_n_i_t_y}[/tex][tex]\frac{a_n_+_1}{a_n}[/tex][tex]\pm[/tex] [tex]\frac{1}{a_n*a_n_-_1}[/tex]}

Lim[tex]\{_n_-_>_i_n_f_i_n_i_t_y}[/tex][tex]\pm[/tex] [tex]\frac{1}{a_n*a_n_-_1}[/tex]} = 0

in the limit [tex]\frac{a_n}{a_n_-_1}[/tex] = [tex]\frac{a_n_+_1}{a_n}[/tex]

Why this do not prove the convergence?

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