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We have sequences $a_n$ such that:

  • $a_1=1, a_2=2,$
  • $a_{n+k}=a_n$ - for some unknown $k \in \mathbb{N}$. $n=1,2,3,...$

And for the sequences $b_n=a_{n+2}-a_{n+1}+a_n$ we know that:

$ b_{n+1}=\frac{1}{2}(b_n^2+1). $

Find $a_n$ - ?

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    I mean general formula for $a_n$. $a_3 -?, a_4 -?$ and so on...2012-11-14

1 Answers 1

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One answer is for $a_n$ to be the sequence $1,2,2,1,0,0$ repeating.

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    Ok. Here my thoughts. If b_n > b_{n+1}, then b_n>(1+b_n^2)/2, and (b(n)-1)^2 <0. So, we have that $b_n$ is periodic, nondecreasing, therefore $b_n$ is a sequence of constants. It's possible only if all $b_n =1$. So we have that $b_1=1$. And the only answer is $a_n= (1,2,2,1,0,0,...)$ repeating.2012-11-15