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Math Help - Help with another succesion problem (Fixed)

  1. #1
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    Help with another succesion problem (Fixed)

    Succesion \{F_n\}\  n\geq0 is defined  F_0=0,  F_1=1,  f_{n+2}=f_{n+1}+f_{n} (fibonacci). For each  n \epsilon N (what's the command for natural numbers?)

    Show for all m\geq0:<br />
\sum_{k=0}^{m}F_k^2 = F_m * F_{m+1}.

    This is what I have so far:

    Induction plugging in 0 works.

    \sum_{k=0}^{m}F_k^2 + (m+1)^2 = F_m*F_{m+1} + (m+1)^2 = F_{m+1}*F_{m+2}

    F_{m+1} * F_{m+2} = F_{m+1}*(F_{m+1} + F_m) = F_m*F_{m+1} + F_{m+1}*F_{m+1}

    As far as I got.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by HeadOnAPike View Post
    Succesion \{F_n\}\  n\geq0 is defined  F_0=0,  F_1=1,  f_{n+2}=f_{n+1}+f_{n} (fibonacci). For each  n \epsilon N (what's the command for natural numbers?)

    Show for all m\geq0:<br />
\sum_{k=0}^{m}F_k^2 = F_m * F_{m+1}.

    This is what I have so far:

    Induction plugging in 0 works.

    \sum_{k=0}^{m}F_k^2 + (m+1)^2 = F_m*F_{m+1} + (m+1)^2 = F_{m+1}*F_{m+2}
    Why are you adding (m+1)^2?
    \sum_{k=0}^{m+1}F_k^2, which is what you want for a proof by induction, is \sum_{k=0}^m F_k^2+ F_{m+1}^2

    F_{m+1} * F_{m+2} = F_{m+1}*(F_{m+1} + F_m) = F_m*F_{m+1} + F_{m+1}*F_{m+1}

    As far as I got.
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