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Math Help - Recurrence Relation Problem...

  1. #1
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    Recurrence Relation Problem...


    Solve the recurrence relation :
    an
    = 2an−1 2an−2
    subject to the initial conditions

    a0 = 1, a1 = 3.
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  2. #2
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    First solve the characteristic equation s^2-2s+2= 0

    What do you get?
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  3. #3
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    Quote Originally Posted by pickslides View Post
    First solve the characteristic equation s^2-2s+2= 0

    What do you get?
    Imaginary roots,

    1+1i , 1-1i
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  4. #4
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    That sounds right.

    The next step is to put these solutions into polar form. Once you have done this, the general solution to the recurrance relation will be of the form

    \displaystyle a_n = r^n(\alpha \cos n\theta +\beta \sin n\theta)

    using a_0=1,a_1=3 to solve for \alpha and \beta
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  5. #5
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    Quote Originally Posted by pickslides View Post
    That sounds right.

    The next step is to put these solutions into polar form. Once you have done this, the general solution to the recurrance relation will be of the form

    \displaystyle a_n = r^n(\alpha \cos n\theta +\beta \sin n\theta)

    using a_0=1,a_1=3 to solve for \alpha and \beta

    Thanks a lot.

    But, after converting the 1+1i and 1-1i into polar forms, how do I get the achieve that form of equation. Also , in the end , am I likely to end up with a simultaneous equation while solving for the two variables.
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  6. #6
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    Quote Originally Posted by aamiri View Post
    Thanks a lot.

    But, after converting the 1+1i and 1-1i into polar forms, how do I get the achieve that form of equation.
    That equation is the general form of the solution. Just substitute the values you find for r and \theta into it.


    Quote Originally Posted by aamiri View Post
    am I likely to end up with a simultaneous equation while solving for the two variables.
    Yes
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