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Math Help - Sequences

  1. #1
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    Sequences

    Given the sequence:

    u0=3 u1=22

    un+2= (12un+1)-(20un)

    To find the closed for we use the equation

    r^2-12r+20=0

    So, why do we use that equation, how does it work? Why are those signs switched, so positive becomes a negative and vice versa?

    I know the answer and how to get it, so

    un=2^n+2x10^10

    I just need some help on how all this works.

    Thanks.
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  2. #2
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    Quote Originally Posted by mark090480 View Post
    Given the sequence:

    u0=3 u1=22

    un+2= (12un+1)-(20un)

    To find the closed for we use the equation

    r^2-12r+20=0

    So, why do we use that equation, how does it work? Why are those signs switched, so positive becomes a negative and vice versa?

    I know the answer and how to get it, so

    un=2^n+2x10^10

    I just need some help on how all this works.

    Thanks.
    Hi Mark,

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

    then since they are equal, subtract them and the answer is zero.

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

    then write the characteristic equation and get the roots \alpha and \beta

    using U_{n+2}\rightarrow\ r^2,\ U_{n+1}\rightarrow\ r^1,\ U_n\rightarrow\ r^0=1

    r^2-12r+20=(r-10)(r-2)=0

    \alpha=10,\ \beta=2

    Then you use your values of U_1 and U_0 to finish up.
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  3. #3
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    Hello, mark090480!

    Here's a primitive explanation . . .


    Given the sequence: . U_{n+2} \:=\:12\,U_{n+1} - 20\,U_n,\;\;\;U_0 = 3,\;U_1 = 22

    To find the closed for we use the equation . r^2-12r + 20 \:=\:0

    So why do we use that equation, how does it work?

    We have: . U_{n+2} \:=\:12\,U_{n+1} - 20\,U_n


    Assume that U_n is an exponential function: . U_n \:=\:r^n


    Then we have: . r^{n+2} \:=\:12r^{n+1} - 20r^n  \quad\Rightarrow\quad r^{n+2} - 12r^{n+1} + 20r^n \:=\:0


    Divide by r^n\!:\;\;\;r^2 - 12r + 20 \:=\:0


    Got it?

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  4. #4
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    OK thanks, I get it I think.

    so, if we assume the Fibonacci sequence

    5= 3+2

    5-3-2=0

    hence that is why. I'm not sure still as to how solving that equation gives us the closed form, could you explain that?
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