Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: √ ∞ ≠ ≤ ≥ ≈ ⇒ ∈ Δ θ ∴ ∑ ∫ π -

Login

Username

Password

Not registered yet?

#1 2008-03-20 07:07:23

tony123
Full Member

Offline

Determine all real x

Determine all real x

such that




is an integer for any integer n.

 

#2 2008-03-20 08:07:15

JaneFairfax
Legendary Member

Offline

Re: Determine all real x

Sorry

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

Last edited by JaneFairfax (2008-03-20 08:20:34)


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

A: Click here for answer.
 

#3 2008-03-24 00:43:35

Kurre
Power Member

Offline

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 (2008-03-24 00:45:43)

 

#4 2008-03-24 01:00:40

Daniel123
Power Member

Offline

Re: Determine all real x

 

#5 2008-03-24 01:06:43

Kurre
Power Member

Offline

Re: Determine all real x

Thanks! smile

 

Board footer

Powered by FluxBB