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.