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Mathematics-Online problems:

Problem of the week

For a sequence $ x_1, \ x_2, \ldots $ let each term be $ 1$ less than the sum of its two neighbors:

$\displaystyle x_k = x_{k-1} + x_{k+1}-1, \quad k = 2, 3, \ldots . $

Every such sequence is periodic.

Hint:
A sequence is called periodic if $ x_{k+n} = x_k$ for $ k = 1, 2, \ldots$ and some $ n \in \mathbb{N}$. The smallest $ n$ having this property is called the length of the period of that sequence.


Answer:

Length of the period:

 
Sum:

Terms of sequence:		  
$ x_{11} =$
$ x_{22} = $
$ x_{33} = $
$ x_{44} = $
$ x_{55} = $


   

[solution to the problem of the previous week]