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Here's the question: The immigration rate to the Czech republic is currently $77000$ peeople per year. Because of a low fertility rate, the population is shrinking at a continuous rate of $0.1$% per year. The current Czech population is ten million.

Assume the immigrants immediately adopt the fertility rate of their new country. If this scenario did not change, would there be a terminal population projected for the Czech republic? If so, find it.

I need to establish a limit as time would approach infinity (terminal population) but the only way Ive been able to solve it has been using $(x_{n-1} + 77000)(0.999) = x_n$ and I'm not sure exactly how to establish a limit off that (can't use differential equations, we never learned this year). Is there another way to solve for the terminal population without using differential equations?

I've tried multiple ways to get a better answer, but the prof said we need to use limits and I'm stumped.

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What you have there is not a differential equation; it's a plain old difference equation or recursively defined sequence. In order to be a differential equation it would have to contain the derivative of the unknown function somewhere. – Henning Makholm Dec 6 '11 at 2:52

We will proceed in two steps: First, assuming that the limit $\lim \limits_{n \to \infty} x_n$ exists, we will find it. Of course, we need to justify our assumption. So we will come back and show the existence of the limit.

Finding the limit. Suppose $x = \lim \limits_{n \to \infty} x_n$. Then allowing $n$ to go to infinity in $$ x_n = 0.999(x_{n-1} + 77000), $$ we get $$ x = 0.999 x + 76992.3 \qquad \implies x = \cdots. $$ (Exercise: Fill in the blank!)

Showing the existence of the limit. We suspect that for large $n$, we must have $x_n$ approaching $x = \cdots$, so we will think about the "corrected" sequence $x_n - x$ instead. That is, subtracting $x$ from both sides of the equation $$ x_n = 0.999(x_{n-1} + 77000), $$ we get $$ x_n - x = 0.999(x_{n-1} + 77000) - x \stackrel{\color{Red}{(!!)}}{=} 0.999(x_{n-1} - x). $$ Be sure to check the step marked with $\color{Red}{(!!)}$.

So we end up with $$ (x_n - x) = 0.999(x_{n-1} - x). $$ Making the substitution $x_n - x = y_n$ (for all $n$), we get $$ y_n = 0.999 y_{n-1}. $$ Can you take it from here?

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Just having some trouble wondering what is happening at the (!!), unsure how that -x moves into the brackets – Donoyaaa Dec 6 '11 at 3:26
@Donoyaaa What do you get when you expand the two sides and check whether the two sides are equal? The left hand side is $0.999 x_{n-1}+ 77000 \cdot 0.999 - x$. The right hand side is $0.999x_{n-1} - 0.999 x$. You know the numerical value of $x$, so just plug in and check it. – Srivatsan Dec 6 '11 at 3:29

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