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Math Help - Fibonacci

  1. #1
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    Smile Fibonacci

    { a_0,a_1,a_2,....,a_n,....} := {1,1,2,3,5,8,13,21,34,...}

    What is a generic term a_n in this sequence using a 2 X 2 matrix??

    What is a formula for a_nthat only depends on n, where n ≥ 3 is a positive integer ???

    Any idea for this? I don't know how can I start this??

    Thanks.
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  2. #2
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    Hello, noppawit!

    a_0,a_1,a_2, \hdots ,a_n, \hdots \;=\;\{1,1,2,3,5,8,13,21,34,\hdots\}

    What is a generic term a_n in this sequence using a 22 matrix?
    . . Sorry, I don't understand this.

    What is a formula for a_n that only depends on n, where n ≥ 3 is a positive integer ?

    What course are you in?

    The closed form for the Fibonacci Sequence is: . a_n \;=\;\frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n} {2^n\sqrt{5}}

    Its derivation requires rigorous algebra.

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  3. #3
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    Hello noppawit
    Quote Originally Posted by noppawit View Post
    { a_0,a_1,a_2,....,a_n,....} := {1,1,2,3,5,8,13,21,34,...}

    What is a generic term a_n in this sequence using a 2 X 2 matrix??

    What is a formula for a_nthat only depends on n, where n ≥ 3 is a positive integer ???

    Any idea for this? I don't know how can I start this??

    Thanks.
    There's a whole lot on the Fibonacci sequence, including a matrix form, on Wikipedia here: here

    Grandad
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  4. #4
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    Quote Originally Posted by noppawit View Post
    { a_0,a_1,a_2,....,a_n,....} := {1,1,2,3,5,8,13,21,34,...}

    What is a generic term a_n in this sequence using a 2 X 2 matrix??
    ...
    Check the online encyclopedia of integer sequences.

    Fibonacci numbers:
    F(0)= 0, F(1)= 1, F(2)= 1, F(3)= 2, F(4)= 3


    The matrix representation

    \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n =<br />
\begin{bmatrix} F_{n+1} & F_n \\<br />
F_n & F_{n-1} \end{bmatrix}

    or this
     \begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^n \, = \, \begin{bmatrix} F_{n-1} & F_n \\ F_n & F_{n+1} \\ \end{bmatrix}



     \begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^4 \, = \, \begin{bmatrix} 2 & 3 \\ 3 & 5 \\ \end{bmatrix}


     \begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^9 \, = \, \begin{bmatrix} 21 & 34 \\ 34 & 55 \\ \end{bmatrix}


     \begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^{20} \, = \, \begin{bmatrix} 4181 & 6765 \\ 6765 & 10946 \\ \end{bmatrix}


     \begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^{30} \, = \, \begin{bmatrix} 514229 & 832040 \\ 832040 & 1346269 \\ \end{bmatrix}


    .
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