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Let $u_n$ be a sequence defined on natural numbers (the first term is $u_0$) and the terms are natural numbers ($u_n\in \mathbb{N}$ )

We defined the following sequences:

$\displaystyle \large x_n=u_{u_n}$ $\displaystyle \large y_n=u_{u_n}+1$

If we know that both $y_n$ and $x_n $ are arithmetic sequences ,how we can prove that $u_n $ is also arithmetic sequence

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    As written, the fact that $\{y_n\}$ is arithmetic is redundant given that $\{x_n\}$ is. Did you mean: $\displaystyle \large x_n=u_{u_n}$ AND either $\displaystyle \large y_n=x_{u_n}+1$ or $\displaystyle \large y_n=u_{x_n}+1$?2015-07-09

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