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#1 2008-03-19 08:07:23

tony123
Member
Registered: 2007-08-03
Posts: 189

Determine all real x

Determine all real x

such that


is an integer for any integer n.

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#2 2008-03-19 09:07:15

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Determine all real x

Sorry …

I thought I got x = ±1, but my working was wrong.

Last edited by JaneFairfax (2008-03-19 09:20:34)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#3 2008-03-23 01:43:35

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Determine all real x

assume that x+1/x is an integer. Squaring gives:


which also is an integer
now assume that

are both integers
the following product is an integer:

therefor, x^(n+1) + x^(-n-1) is an integer and by the induction principle it holds for all n.
Hence all x that satisifies the condition is the ones that satisfy that x+1/x is an integer, which implies that there exists infinite many reals satisfying the problems condition.

ps. how do i do the sign, that shows that an element belongs to a set in mathcode?

Last edited by Kurre (2008-03-23 01:45:43)

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#4 2008-03-23 02:00:40

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Determine all real x

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#5 2008-03-23 02:06:43

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Determine all real x

Thanks! smile

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