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PEN Math problems from the collection "Problems in Elementary Number Theory"

Math problems from the collection "Problems in Elementary Number Theory"

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PEN Math problems from the collection "Problems in Elementary Number Theory"

Math problems from the collection "Problems in Elementary Number Theory"

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Recursive Sequences

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Peter

#1 h May 25, 2007, 12:25 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

The sequence $\{x_{n}\}_{n \ge 1}$ is defined by
$x_{1} = 2, x_{n + 1} = \frac {2 + x_{n}}{1 - 2x_{n}}\;\; (n \in \mathbb{N}).$ Prove that

a) $x_{n}\not = 0$ for all $n \in \mathbb{N}$ ,

b) $\{x_{n}\}_{n \ge 1}$ is not periodic.

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TTsphn

#2 h Oct 21, 2007, 1:23 AM • 2 Y

Y by Adventure10 and 1 other user

Peter wrote:

The sequence $\{x_{n}\}_{n \ge 1}$ is defined by
$x_{1} = 2, x_{n + 1} = \frac {2 + x_{n}}{1 - 2x_{n}}\;\; (n \in \mathbb{N}).$
Prove that

$x_{n}\not = 0$ for all $n \in \mathbb{N}$ ,
$\{x_{n}\}_{n \ge 1}$ is not periodic.

$a = \arctan {2}$
$x_1 = \tan{a}$
$x_{n + 1} = \frac {x_n + \tan{a}}{1 - x_n\tan{a}}$
We prove by induction that:
$x_{n} = \tan{na}$
Suppose $x_n$ is period T.
$\tan{(n + T)a} = \tan{na}$
$\rightarrow Ta\equiv 0(\mod \pi)$
So $a = \frac {mk\pi}{T}$
We has a result :
$k\in Q$ then $\tan{k\pi}$ is not equal 2.

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#3 h Apr 15, 2008, 1:58 AM • 2 Y

Y by Adventure10, Mango247

Another approach

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