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Infinite Series Word Problem

  1. Dec 18, 2016 #1
    1. The problem statement, all variables and given/known data

    A fishery manager knows that her fish population naturally increases at a rate of 1.4% per month, while 119
    fish are harvested each month. Let Fn be the fish population after the nth month, where F0 = 4500 fish. Assume that that process continues indefinitely. Use the infinite series to find the long-term (steady-state) population of the fish exactly.

    2. Relevant equations


    3. The attempt at a solution

    My issue is that I can't seem to set up an expression to evaluate the series. I know that the expression will involve subtracting 119 and use 0.014 to represent the percent increase. If it were only the percent increase, I would be able to set up an expression. But the -119 is really throwing me off.
     
  2. jcsd
  3. Dec 18, 2016 #2

    haruspex

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    The question does not make clear whether the Fn represent the population just after a harvest or just before. I would take it as just after.
    If the population is Fn after the nth month what will it be after one more month?
     
  4. Dec 18, 2016 #3
    Fn+1 = Fn(1 + 0.014) - 119?
     
  5. Dec 18, 2016 #4

    haruspex

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    Right. Do you know a way to solve such equations? If not, an easy thing to try is to see if you can add a constant to each Fn so that it reduces to a simple geometric progression.
     
  6. Dec 18, 2016 #5
    I don't know how to write this equation without Fn being in terms of Fn+1 or Fn-1.
     
  7. Dec 18, 2016 #6

    haruspex

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    It will be the same equation, but written in the form (Fn+1+c)=a(Fn+c) for some pair of constants a and c.
     
  8. Dec 18, 2016 #7
    How will adding the c to both sides eliminate the Fn+1? None of the problems I have done or have examples of with infinite series so far have anything like this, so I don't really have anything to go by.
     
  9. Dec 18, 2016 #8

    haruspex

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    I did not suggest it would.
    You have this equation: Fn+1 = Fn(1 + 0.014) - 119
    and I am suggesting this form of it: (Fn+1+c)=a(Fn+c)
    What do you get if you combine them?
     
  10. Dec 18, 2016 #9

    Ray Vickson

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    You have ##F_{n+1} = 1.014 F_n - 119## with ##F_0 = 4500##. Try calculating ##F_1, F_2, F_3## (keeping ##F_0## symbolic instead of 4500). In fact, it might make everything much clearer if you keep all parameters symbolic, so that ##F_{n+1} = r F_n - k##. Using symbols like that instead of numbers helps keep separate the different effects.

    However, I think there is something very wrong with the original problem statement: for ##r > 1## (for example, for ##r = 1.014##) you must have a very special relationship between ##F_0,r,k## in order to obtain a finite limit; otherwise you will either have ##F_n \to +\infty## as ##n \to \infty## (for some combinations of ##F_0##, ##r##, and ##k##) or else ##F_n \to -\infty## for for other combinations. Of course, the latter case really means that ##F_n## hits zero at some finite ##n## and so the fish population dies out completely and the problem ends; ##F_n## does not actually go to ##-\infty##.
     
  11. Dec 18, 2016 #10
    By combine them, do you mean take a Fn+1 in the second equation as Fn+1 = Fn(1+0.014) - 119 and then try to solve for a and C?
     
  12. Dec 18, 2016 #11

    haruspex

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    I assumed it was intended that:
     
  13. Dec 18, 2016 #12

    haruspex

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    Yes. You will have one equation with two unknowns, but remember that the equation has to be true for all Fn.
     
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