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Thread: Solve the equation

  1. #1
    Junior Member
    May 2009

    Solve the equation

    Obtain the sol'n for an=5an-2 - 4an-4

    a0 = 3
    a1 = 2
    a2 = 6
    a3 = 8

    The 5an-2 minus 4an-4 actually appear in the the problem as 5a and 4a with the terms being n-2 and n-4 list slightly lower than the an term
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  2. #2
    MHF Contributor red_dog's Avatar
    Jun 2007
    Medgidia, Romania
    $\displaystyle a_n-5a_{n-2}+4a_{n-4}=0$

    The characteristic equation is $\displaystyle t^4-5t^2+4=0$ with the roots

    $\displaystyle t_1=1, \ t_2=-1, \ t_3=2, \ t_4=-2$

    We have to find the general term $\displaystyle a_n$ in the form $\displaystyle a_n=a+b(-1)^n+c\cdot 2^n+d(-2)^n$

    For $\displaystyle n=0, \ 1, \ 2, \ 3$ we have

    $\displaystyle a_0=a+b+c+d=3$

    $\displaystyle a_1=a-b+2c-2d=2$

    $\displaystyle a_2=a+b+4c+4d=6$

    $\displaystyle a_3=a-b+8c-8d=8$

    Now find a, b, c, d.
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