Prove L_n = L_n-1 + L_n-2 for n >= 3, where L_n are Lucas numbers.

Thank you!

- Jan 9th 2007, 10:59 PMyc6489Prove L_n = L_n-1 + L_n-2 for n >= 3, where L_n are Lucas numbers.
Prove L_n = L_n-1 + L_n-2 for n >= 3, where L_n are Lucas numbers.

Thank you! - Jan 10th 2007, 04:13 AMSoroban
Hello, yc6489!

Sorry, I don't understand the question . . .

Quote:

Prove $\displaystyle L_n \:= \:L_{n-1} + L_{n-2}$ for $\displaystyle n \geq 3$, where $\displaystyle L_n$ are Lucas numbers.

But that is the**definition**of Lucas Numbers.

. . $\displaystyle L_n\:=\:L_{n-1} + L_{n-2}$ where $\displaystyle L_1 = 1,\:L_2 = 2$

So what is there to prove?

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It's similar to saying:

"Prove that $\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$ is the Harmonic Series."

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Fibonacci started with 1 and 1.

Lucas started with 1 and 2 ... and gets into the history books.

Okay . . . $\displaystyle S_1 = 1,\:S_2 = 4$

. . . . . . .$\displaystyle S_n\:=\:S_{n-1} + S_{n-2}$ for $\displaystyle n \geq 3.$

We have the sequence: .$\displaystyle 1,\,4,\,5,\,9,\,14,\,23,\,\hdots$

. . which are the lesser-known Soroban numbers.

- Jan 10th 2007, 04:42 AMCaptainBlack
What you are asked to prove is the usual definition of the Lucas numbers (when you add in

the initial valuse L_0=2, L_1=1, but we can use an alternative definition and prove the

usual one from it.

Take as the definition of Lucas number L_n, n>=1, (and without restiction if we extend the

Fibonacci numbers to negative index):

L_n=F_(n-1)+F_(n+1),

then if n>=3:

L_n=[F_(n-3)+F_(n-2)]+[F_(n-1)+F_(n)]

.....=[F_(n-3)+F_(n-1)]+[F_(n-2)+F_(n)]

.....=L_(n-2)+L_(n-1)

RonL - Jan 10th 2007, 05:34 AMThePerfectHackerQuote:

Fibonacci started with 1 and 1.

Lucas started with 1 and 2 ... and gets into the history books.

*)But it has happened. The Pell equation is named after someone who did nothing with it, rather then the actual inventor, Fermat.

*)The L'Hopital rule is named after Guillalame de L'Hopital (I wish my name was that) because he wrote the first Calculus book ever and it featured it. While the actual discoverer was Johann Bernouilli.