# Iterating the transform $(a,b)\mapsto(a+b+\sqrt{a^2+b^2} ,a+b-\sqrt{a^2+b^2})$

Assume that $a_0=-2$, $b_0=1$, and that, for every $n\ge0$, $$a_{n+1}=a_n+b_n+\sqrt{a^2_n+b^2_n} \qquad b_{n+1}=a_n+b_n-\sqrt{a^2_n+b^2_n}$$

How to find $a_{2012}$?

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Why didn't Math.StackExchange start 2000 years earlier? It would make life much more easier... – draks ... Jul 19 '12 at 14:50
Hint: What is $a_{n+1}+b_{n+1}$, and what is $a_{n+1}^2+b_{n+1}^2$? – celtschk Jul 19 '12 at 14:59
Got something from an answer below? – Did Jul 25 '12 at 16:29
Bis repetita placent. – Did Jul 29 '12 at 13:50
@did I added them and multiplied them ,but what is the next step ? – Frank Jul 29 '12 at 15:44

Here is a 7-steps plan:

1. Stop asking questions with no indication whatsoever about which similar problems you can solve, what you tried to solve the present one, and why you think your attempts failed.

3. Define $s_n=a_n+b_n$ and $p_n=a_nb_n$ and, for every $n\geqslant0$, express $s_{n+1}$ and $p_{n+1}$ in terms of $s_n$ and $p_n$.
4. Compute $s_0$ and $p_0$.
5. For every $n\geqslant1$, express $a_n$ and $b_n$ in terms of $s_n$ and $p_n$.