Results 1 to 4 of 4

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

  1. #1
    Newbie
    Joined
    Oct 2006
    Posts
    4

    Prove 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!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,707
    Thanks
    626
    Hello, yc6489!

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


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

    But that is the definition of Lucas Numbers.

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


    So what is there to prove?

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

    It's similar to saying:

    "Prove that 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 . . . S_1 = 1,\:S_2 = 4
    . . . . . . . S_n\:=\:S_{n-1} + S_{n-2} for  n \geq 3.


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

    . . which are the lesser-known Soroban numbers.

    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by yc6489 View Post
    Prove L_n = L_n-1 + L_n-2 for n >= 3, where L_n are Lucas numbers.

    Thank you!
    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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Fibonacci started with 1 and 1.
    Lucas started with 1 and 2 ... and gets into the history books.
    I think he acomplished something, they would just not place his name without any good reason.*



    *)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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Lucas Sequences
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: May 21st 2010, 06:21 AM
  2. Proof about Fibonacci and Lucas numbers (GCD)
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: April 21st 2010, 09:26 AM
  3. help with Lucas numbers
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 2nd 2009, 12:32 AM
  4. Lucas Numbers
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 4th 2009, 04:27 AM
  5. Proving Lucas Numbers and Fibonacci Numbers
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 18th 2008, 11:33 PM

Search Tags


/mathhelpforum @mathhelpforum