I am sorry if this is a very stupid question, but I am new in the topic and I was not able to find either a solution or relevant literature. (I am so new that I really don't know even what the right tags are!)


I have a system of two linear recurrences. In particular, the first recurrence is $x\_n=998x\_{n-1}+995y\_{n-1}$; the second recurrence is $y\_n=1000x\_{n-1}+996y\_{n-1}$. The initial conditions are $x\_1=\frac{1993}{2}$ and $y\_1=998$.


Can one write $x\_n$ and $y\_n$ in closed form?


More specifically, I am interested in understanding the limit of $\frac{x\_n}{x\_n+y\_n}$. It sound like being very close to $\frac{1}{2}$. Related to this, does $\frac{x\_n}{y\_n}$ converges to 1?


Thank you in advance for any help.


Valerio



A system of linear recurrences
The limit of $x\_n/(x\_n+y\_n)$ is not 1/2 but


$$331667 + 332\sqrt{995001}\over 663334 + 665\sqrt{995001}$$


The limit of $x\_n/y\_n$ is not 1 but 


$$331667 + 332\sqrt{995001}\over 331667 + 333\sqrt{995001}$$


