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Math Help - Fibonacci number

  1. #1
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    Fibonacci number

    For  n \in N let  f_n be the n^{th} Fibonacci number :
     f_1 = 1 \ , f_2 =1 , f_n= f_{n-1}+ f_{n-2} , n \geq 3 .
    Prove that :
    f_n \ is \ even \iff 3 \ divides \ n
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  2. #2
    MHF Contributor chisigma's Avatar
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    According to...

    Fibonacci number - Wikipedia, the free encyclopedia

    ... a basic property of the fibonacci's numbers is that 'every 3rd number of the sequence is even and more generally, every k-th number of the sequence is a multiple of f_{k}'. In case of k=3 is f_{k}=3...

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by flower3 View Post
    For  n \in N let  f_n be the n^{th} Fibonacci number :
     f_1 = 1 \ , f_2 =1 , f_n= f_{n-1}+ f_{n-2} , n \geq 3 .
    Prove that :
    f_n \ is \ even \iff 3 \ divides \ n
    Proof by induction:

    Let the statement be P(n).

    f_3=1+1=2

    Thus P(3) is true.

    --------------------------------------------------

    Let P(m) be true.

    m=3a

    f_m=2k

    --------------------------------------------------

    Consider P(m+3).

    m+3=3a+3=3(a+3)

    Thus 3 divides m+3.

    f_{m+3}=f_{m+2}+f_{m+1}

    f_{m+3}=f_{m+1}+f_m+f_{m+1}

    f_{m+3}=2f_{m+1}+f_m

    f_{m+3}=2f_{m+1}+2k

    f_{m+3}=2(f_{m+1}+k)

    Thus f_{m+3} is even.

    --------------------------------------------------

    Hence, by the principle of induction, the statement is true.
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