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Thread: 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 $\displaystyle U_{n+2}=12U_{n+1}-20U_n$

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

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

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

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

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

    $\displaystyle \alpha=10,\ \beta=2$

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

    Here's a primitive explanation . . .


    Given the sequence: .$\displaystyle 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 .$\displaystyle r^2-12r + 20 \:=\:0$

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

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


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


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


    Divide by $\displaystyle 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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