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Math Help - Recursive sequence limit challenge

  1. #1
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    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
    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 { G }_{ n }=\frac { { F }_{ n+1 } }{ { F }_{ n } } an if we took the limit \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:

    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 \lim _{ n\rightarrow \infty  }{ \frac { { T }_{ n+1 } }{ { T }_{ n } }  } ?

    A lot of thanks and
    Regards,
    ManuelSG
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  2. #2
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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

    Recursive sequence limit challenge-mhfcalc22.png
    Thanks from ManuelSG
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