# Thread: Help with another succesion problem (Fixed)

1. ## 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.

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.