# Thread: Help with understanding the proof of Lucas' formula (strong version of induction)

1. ## Help with understanding the proof of Lucas' formula (strong version of induction)

Hi.
I don't understand the proof of Lucas' formula. I would be very happy is someone could explain it step by step. Especially the last to lines of the proof. Here is a link to the proof.

- Gøran

2. ## Re: Help with understanding the proof of Lucas' formula (strong version of inductio

Well, it's too much work to expand on the whole page of a proof with complicated formulas. If you have a smaller concrete question, I would be happy to help.

Especially the last to lines of the proof.
Do you mean

$\displaystyle =F_k+F_{k+1}$
$\displaystyle =F_{k+2}$ ?

See this page for a more intuitive proof.

3. ## Re: Help with understanding the proof of Lucas' formula (strong version of inductio

Hi,
What theorem in the link (12.4, 12.5 or 12.6) do you not understand? The first is just an induction and the next two just use the fact that
$\displaystyle F_n={\alpha^n-\beta^n\over\alpha-\beta}$ with $\displaystyle \alpha={1+\sqrt5\over2$ and $\displaystyle \beta={1-\sqrt5\over2}$