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Thread: Help with another succesion problem (Fixed)

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

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

    Show for all $\displaystyle m\geq0:
    \sum_{k=0}^{m}F_k^2 = F_m * F_{m+1}.$

    This is what I have so far:

    Induction plugging in 0 works.

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

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

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

    Show for all $\displaystyle m\geq0:
    \sum_{k=0}^{m}F_k^2 = F_m * F_{m+1}.$

    This is what I have so far:

    Induction plugging in 0 works.

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

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

    As far as I got.
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