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Math Help - finding the closed for of this sequence

  1. #1
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    finding the closed for of this sequence

    xn=30

    xn+1 = 1.8xn +15

    I'm completely lost with this question, finding the closed form of the linear recurrence sequence

    any help would be appreciated, thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by entrepreneurforum.co.uk View Post
    xn=30

    xn+1 = 1.8xn +15

    I'm completely lost with this question, finding the closed form of the linear recurrence sequence

    any help would be appreciated, thanks

    The statement has no sense. Have you exactly quoted it? .


    Fernando Revilla
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  3. #3
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    X1= 30 sorry
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  4. #4
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    Hello, entrepreneurforum.co.uk!

    X_{n+1} \:=\:1.8X_n + 15,\;\;X_1 = 30

    Find the closed form of the recurrence sequence.

    [rant]
    . . . . . Why are these problems assigned to people
    . . . . . who have no knowledge of recurrence relations?
    [/rant]


    \begin{array}{ccccccc}\text{We have the recurrence: } & X_{n+1} &=& 1.8X_n + 15 & [1] \\<br />
\text{Write the "next" equation:} & X_{n+2} &=& 1.8X_{n+1} + 15 & [2] \end{array}

    Subtract [2] - [1]: . X_{n+2} - X_{n+1} \;=\;1.8X_{n+1} - 1.8X_n

    . . . . . X_{n+2} - 2.8X_{n+1} + 1.8X_n \;=\;0


    \text{Let }Y^n = X_n\!: \;\;Y^{n+2} - 2.8Y^{n+1} + 1.8Y^n \;=\;0

    \text{Divide by }Y^n\!:\;\;Y^2 - 2.8Y + 1.8 \;=\;0 \quad\Rightarrow\quad (Y - 1)(Y - 1.8)\;=\;0

    . . Hence: . Y = 1,\;Y = 1.8

    The generating function is of the form: . f(n) \;=\;A + B(1.8)^n


    We know the first two terms of the sequence: . f(1) = 30,\;f(2) = 69

    . . \begin{array}{ccccccc} f(1) = 30\!: & A + 1.8B\; &=& 30 & [3] \\ f(2) = 69\!: & A + 3.24B &=& 69 & [4] \end{array}

    Subtact [4] - [3]: . 1.44B \:=\:39 \quad\Rightarrow\quad B \:=\:\dfrac{39}{1.44} \:=\:\dfrac{325}{12}

    Substitute into [3]: . A + 1.8\left(\frac{325}{12}\right) \:=\:30 \quad\Rightarrow\quad A \:=\:\text{-}\dfrac{75}{4}


    Therefore: . f(n) \;=\;-\dfrac{75}{4} + \dfrac{325}{12}(1.8)^n


    . . . . . . . . f(n) \;=\;\dfrac{25}{12}\bigg[13(1.8)^n - 9\bigg]

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