Help with another succesion problem (Fixed)

**HeadOnAPike** · October 12th 2009, 04:53 PM

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.

**HallsofIvy** · October 12th 2009, 07:20 PM

Originally Posted by HeadOnAPike

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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