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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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    Cool. $ $ $ $ $ $2012-09-20

1 Answers 1

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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.

  2. Stop ignoring comments asking you to add the pieces of information mentioned in 1.

  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 $p_n$.

  6. Conclude.

  7. Memorize 1. and 2. for your next questions on the site.