# Infinite Series Word Problem

1. Dec 18, 2016

### Burjam

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. Dec 18, 2016

### haruspex

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?

3. Dec 18, 2016

### Burjam

Fn+1 = Fn(1 + 0.014) - 119?

4. Dec 18, 2016

### haruspex

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.

5. Dec 18, 2016

### Burjam

I don't know how to write this equation without Fn being in terms of Fn+1 or Fn-1.

6. Dec 18, 2016

### haruspex

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.

7. Dec 18, 2016

### Burjam

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.

8. Dec 18, 2016

### haruspex

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?

9. Dec 18, 2016

### Ray Vickson

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 $-\infty$.

10. Dec 18, 2016

### Burjam

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?

11. Dec 18, 2016

### haruspex

I assumed it was intended that:

12. Dec 18, 2016

### haruspex

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