Recursive sequence limit challenge

Hi, I saw this problem in Coursera's Calculus II by the Ohio State University:

Most of us are familiar with the Fibonacci Sequence

$\displaystyle With\quad F_{ 0 }={ F }_{ 1 }=1\\ \\ { F }_{ n }={ F }_{ n-1 }+{ F }_{ n-2 }\\ \\ \left\{ 1,1,2,3,5,8,13,21,...{ F }_{ n } \right\} $

Now we can build $\displaystyle { G }_{ n }=\frac { { F }_{ n+1 } }{ { F }_{ n } } $ an if we took the limit $\displaystyle \lim _{ n\rightarrow \infty }{ { G }_{ n } } =\frac { 1+\sqrt { 5 } }{ 2 } =\phi $

First I want to know how to compute that kind of limit, I tried some of the things I have at hand(That aren't very much since I haven't done a googling about sequence limits nor continue with the course) but without results.

The other limit I want you to check is form the so-called Tribonacci sequence:

$\displaystyle With\quad T_{ 0 }={ T }_{ 1 }={ T }_{ 2 }=1\\ \\ { T }_{ n }={ T }_{ n-1 }+{ T }_{ n-2 }+{ T }_{ n-3 }\\ \\ \left\{ 1,1,1,3,5,9,17,31,...{ ,T }_{ n } \right\} $ then, as you can guess, what is the value of $\displaystyle \lim _{ n\rightarrow \infty }{ \frac { { T }_{ n+1 } }{ { T }_{ n } } } $ ?

A lot of thanks and

Regards,

ManuelSG :)

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Re: Recursive sequence limit challenge

Hi,

In order to follow the solution in the attachment, you need to know about linear homogeneous constant coefficient recurrence relations -- Recurrence relation - Wikipedia, the free encyclopedia

Attachment 29627