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Fibonacci sequence

  1. May 20, 2009 #1
    the Fibonacci sequence is defined by

    a1 = a2 = 1, a(n+2)] = an + a(n+1).

    write out the first 6 terms of the sequence and prove that an = 1/[tex]\sqrt{5}[/tex][ ((1+[tex]\sqrt{}5[/tex])/2)^2 - ((1-[tex]\sqrt{}5[/tex])/2)^2]
     
  2. jcsd
  3. May 20, 2009 #2
    the first 6 terms are 1,1,2,3,5,8 but from here where do i go proving this... totally lost
     
  4. May 20, 2009 #3
    wait do i have to use the first 6 terms or do you think they just want me to prove it?
    Thanks for any replies
     
  5. May 20, 2009 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You've got the first 6 terms just fine. To find the general solution look for power law solutions an=r^n of your recursion relation. Put an=r^n into your recursion relation and solve for r. You should get a quadratic and two solutions. What are they?
     
  6. May 20, 2009 #5

    Mark44

    Staff: Mentor


    This can't be the right formula for an. It varies with n, while your formula above is a constant.
     
  7. May 20, 2009 #6
    The characteristic equation is [tex] r^{2}-r-1=0[/tex]

    The roots are [tex] \frac {1+ \sqrt{5}}{2} , \frac {1- \sqrt{5}}{2} [/tex]

    The general solution is then [tex]a_{n} = \alpha (\frac {1+ \sqrt{5}}{2})^{n} + \beta ({\frac{1- \sqrt{5}}{2})^n [/tex]

    Then use the initial conditions, namely a0=1 and a1=1 to find [tex] \alpha [/tex] and [tex] \beta [/tex]
     
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