Let $x_{n+1} = 0.9 x_n + 500$ with $x_0 = 10$. The limit of the sequence, $\lim\limits_{n\to \infty} x_n$, seems to be about 5000 when I try it in a spreadsheet.
How can I prove that the limit converges and that 5000 is correct? Also, is there a way to write an equation for $x_n$ that is non-recursive so that elements of the sequence can be calculated directly?