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Math Help - second order recurrence sequences

  1. #1
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    second order recurrence sequences

    Hello,

    I hope I have posted this in the correct place, if not, sorry.

    Okay, I have just started working through the material, but I am having some problems working out one of the examples, which is given below:

    Un+2 = 12Un+1 - 20Un (n = 0,1,2...)

    We are given
    U0 = 1, U1 = 2

    and are asked to work out U2 to U4.

    But, maybe my head is just blocked, but I can't seem to get the figures. I have looked at the answers at the back of the book, but still no luck.

    So, if anyone could point me in the right direction, I would be very thankful.

    Cheers
    Sean

    PS - Sorry about typing the question out, I haven't got the hang of LaTeX just yet.
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  2. #2
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    Quote Originally Posted by feely View Post
    Hello,

    I hope I have posted this in the correct place, if not, sorry.

    Okay, I have just started working through the material, but I am having some problems working out one of the examples, which is given below:

    Un+2 = 12Un+1 - 20Un (n = 0,1,2...)

    We are given
    U0 = 1, U1 = 2

    and are asked to work out U2 to U4.

    But, maybe my head is just blocked, but I can't seem to get the figures. I have looked at the answers at the back of the book, but still no luck.

    So, if anyone could point me in the right direction, I would be very thankful.

    Cheers
    Sean

    PS - Sorry about typing the question out, I haven't got the hang of LaTeX just yet.
    U_{n+2} = 12U_{n+1} - 20U_n

    U_{n + 2} - 12U_{n + 1} + 20U_n = 0

    Solve the characteristic equation
    m^2 - 12m + 20 = 0 \implies m = 2, 10

    So the general solution is going to be
    U_n = A \cdot 2^n + B \cdot 10^n

    Given your two conditions
    U_0 = 1 \implies A + B = 1

    U_1 = 2 \implies 2A + 10B = 2

    So
    A = 1, ~B = 0

    Thus
    U_n = 2^n

    -Dan
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