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Math Help - Recurrence Relation

  1. #1
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    Recurrence Relation

    Hey guys, just hoping that I could get some help with these 2 questions.

    Solve the recurrence relations for these two questions


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  2. #2
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    Quote Originally Posted by Storm20 View Post
    Hey guys, just hoping that I could get some help with these 2 questions.

    Solve the recurrence relations for these two questions


    I just posted this

    http://www.mathhelpforum.com/math-he...-sequence.html

    - it might help.

    Added note. In the case of repeated roots of the characteristic equation, solutions are of the form

    a_n = \left( c_1 n + c_2 \right) \rho^n
    Last edited by Jester; January 14th 2009 at 06:35 AM. Reason: added note
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  3. #3
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    Yeah I thought it might have had something to do with using the general form, but im not 100 percent sure where or how to use it.!
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  4. #4
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    Quote Originally Posted by Storm20 View Post
    Yeah I thought it might have had something to do with using the general form, but im not 100 percent sure where or how to use it.!
    Let's try the first one. Substitute a_n = c \rho^n into

    a_n - 7 a_{n-1} + 10 a_{n-2} = 0

    gives

    c \rho^n - 7 c \rho^{n-1} + 10 c \rho^{n-2} = 0

    or

    (\rho^2 - 7 \rho + 10) c \rho^{n-2} = 0

    which gives
    (\rho^2 - 7 \rho + 10) = 0 or (\rho - 2) \rho - 5) = 0 so \rho = 2,\; 5

    So, the general solution to the difference equation is

    a_n = c_1 2 ^n + c_2 5^n

    Now the initial condition

    a_0 = c_1 + c_2 = -1
    a_1 = c_1 2 + c_2 5 = 4

    Two equations for c_1\; \text{and}\; c_2. SOlving gives

    c_1 = -3,\;\;\;c_2 = 2

    leading to the solution

    a_n = -3 \cdot 2 ^n + 2 \cdot 5^n
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  5. #5
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    Quote Originally Posted by danny arrigo View Post
    Let's try the first one. Substitute a_n = c \rho^n into

    a_n - 7 a_{n-1} + 10 a_{n-2} = 0

    gives

    c \rho^n - 7 c \rho^{n-1} + 10 c \rho^{n-2} = 0

    or

    (\rho^2 - 7 \rho + 10) c \rho^{n-2} = 0

    which gives
    (\rho^2 - 7 \rho + 10) = 0 or (\rho - 2) \rho - 5) = 0 so \rho = 2,\; 5

    So, the general solution to the difference equation is

    a_n = c_1 2 ^n + c_2 5^n
    hmm, I understand how you go to this part, but i'm a little bit confused after that. Your help is really appreciated .

    Quote Originally Posted by danny arrigo View Post
    Now the initial condition

    a_0 = c_1 + c_2 = -1
    a_1 = c_1 2 + c_2 5 = 4

    Two equations for c_1\; \text{and}\; c_2. SOlving gives

    c_1 = -3,\;\;\;c_2 = 2

    leading to the solution

    a_n = -3 \cdot 2 ^n + 2 \cdot 5^n
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