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Math Help - Writing up a Recurrence Relation Problem...

  1. #1
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    Writing up a Recurrence Relation Problem...

    I know how to solve Recurrence Relation problems, I just need help writing up the equation:


    Assume that the deer population of Rustic County is 200 at time
    n = 0 and 220 at time n = 1 and that the increase from time n-1 to time n is twice the increase from time n-2 to time n-1. Write a recurrence relation and an initial condition that de ne the deer population at time n and then solve the recurrence relation.


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  2. #2
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    Let the population be x_n a time n, x_{n-1} at time n-1 and x_{n-2} at time n-2. Using this notation, you know how to write the increase in population from time n-1 to n, don't you?
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  4. #4
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    Hello, aamiri!

    Assume that the deer population of Rustic County is 200 at time 0,
    and 220 at time 1.
    And that the increase from time \,n-1 to time \,n is twice the increase
    from time \,n-2 to time \,n-1.

    (a) Write a recurrence relation and an initial condition
    that define the deer population at time \,n.

    (b) Solve the recurrence relation.

    Let P(t) = deer population at time \,t.

    We are given: . P(0) = 200,\;P(1) = 220


    We are told that: . P(n) \;=\;P(n-1) + 2\!\cdot\!\text{(previous difference)}

    . . . . . . . . . . . . . P(n) \;=\;P(n-1) + 2\!\cdot\![P(n-1) - P(n-2)]

    . . . . . . . . . . . . . P(n) \;=\;P(n-1) + 2\!\cdot\!P(n-1) - 2\!\cdot\!P(n-2)

    . . . . . . . . . . . . . P(n) \;=\;3\!\cdot\!P(n-1) - 2\cdot\!P(n-2) .(a)



    Hence: . P(n) - 3\cdot P(n-1) + 2\cdotP(n-2) \;=\;0

    Let X^n =P(n)\!:\;\;X^n - 3X^{n-1} + 2X^{n-2} \;=\;0

    Divide by X^{n-2}\!:\;\;X^2 - 3X + 2 \:=\:0 \quad\Rightarrow\quad (X-1)(X-2) \:=\:0<br />

    . . Hence: . X \:=\:1,\,2


    The function is of the form: . P(n) \;=\;A\!\cdot1^n + B\!\cdot\!2^n


    We know the first two terms of the sequence:

    \begin{array}{cccccccccc}P(0) = 200\!: & A + B &=& 200 & [1] \\<br />
P(1) = 220\!: & A + 2B &=& 220 & [2] \end{array}

    Subtract [2] - [1]: . B \,=\,20

    Substitute into [1]: . A + 20 \:=\:200 \quad\Rightarrow\quad A \:=\:180


    Therefore: . P(n) \;=\;180 + 20\!\cdot\!2^n (b)



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