# Thread: Fibbonacci and Lucas Number

1. ## Fibbonacci and Lucas Number

Ok. I have made a list of the first 30 lucas and fibbonacci number, and the products they make.

I now need to find a relationship between the products and the fibonacci numbers. I know thats a bit vague but that is what the teacher told us to do, and he said it was blatantly obvious, so either I'm missing something or I've messed up somewhere.

Any help is appreciated.

Thanks.

By the way - I kind of need this answered like immediately!!

2. Originally Posted by Scorpion
Ok. I have made a list of the first 30 lucas and fibbonacci number, and the products they make.

I now need to find a relationship between the products and the fibonacci numbers. I know thats a bit vague but that is what the teacher told us to do, and he said it was blatantly obvious, so either I'm missing something or I've messed up somewhere.

Any help is appreciated.

Thanks.

Learn to do your own research. Go to Wikipedia or MathWorld and
type "lucas numbers" into the search box and then read what it says.

question.

RonL

3. Originally Posted by Scorpion
Ok. I have made a list of the first 30 lucas and fibbonacci number, and the products they make.

I now need to find a relationship between the products and the fibonacci numbers. I know thats a bit vague but that is what the teacher told us to do, and he said it was blatantly obvious, so either I'm missing something or I've messed up somewhere.

Any help is appreciated.

Thanks.

By the way - I kind of need this answered like immediately!!
Hello,

if the elements of the Fibonacci sequence are F_n, n>=0, and
if the elements of the Lucas sequence are L_n, n>=0, and
if the elements of the product sequence are P_n = F_n * L_n, n>=0,

then you can easily detect that:

P_n = 3 * P_{n-1} - P_{n-2}

EDIT: From CaptainBlack's answer I learned that I completely misunderstood your problem. Sorry!

4. Hello, Scorpion!

Ok, I have made a list of the first 30 Lucas and Fibonacci number,
and the products they make.

If you know the generating functions for Fibonacci and Lucas numbers,
. . it's easy . . . well, sort of.

$F_n \;=\;\frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n\sqrt{5}}$

$L_n \;=\;\frac{(1+\sqrt{5})^n + (1-\sqrt{5})^n}{2^n}$

Then: . $F_n\cdot L_n \;=\;\frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^n\sqrt{5}}\cdot \frac{(1 + \sqrt{5})^n + (1-\sqrt{5})^n}{2^n}$

. . . . . . . . . . $= \;\frac{(1+\sqrt{5})^{2n} - (1-\sqrt{5})^{2n}}{2^{2n}\sqrt{5}}\quad\Leftarrow$ . This is $F_{2n}$

Therefore: . $F_n\cdot L_n\;=\;F_{2n}$