Please help me solve a recurrence relation system. I can solve simple recurrence relations, but don’t know how to solve it.
$\begin{cases}x_n = x_{n-1} - \frac{(-1)^n a y_{n-1}}{\sqrt{x_{n-1}^2 + y_{n-1}^2}}\\y_n = y_{n-1} + \frac{(-1)^n a x_{n-1}}{\sqrt{x_{n-1}^2 + y_{n-1}^2}}\end{cases}$
Where $x_0$, $y_0$, $a$ are real numbers.
I also need to find the limit of these sequences such as:
$$\lim_{n\to\infty} \frac{x_n}{\sqrt{x_n^2 + y_n^2}}\\\lim_{n\to\infty} \frac{y_n}{\sqrt{x_n^2 + y_n^2}}$$
Let $z_n = x_n + i y_n$. Then your system can be written as $$ z_{n+1} = z_n (1 + (-1)^n a i/|z_n|)$$ In particular, assuming $a$ is real, $$ |z_{n+1}| = |z_n| \sqrt{1+a^2/|z_n|^2}$$ I'll assume $a \ne 0$ and $z_0 \ne 0$. Then $|z_n|$ is an increasing sequence. If it had a finite limit $L$, then we'd have $L = L \sqrt{1+a^2/L^2}$, and it's easy to see that this has no solution. Thus $|z_n| \to \infty$.