Induction

**Stuck Man** · February 22nd 2013, 03:07 AM

If $x^{3}=x+1$ , prove, by induction or otherwise, that $x^{3n}$ =a_nx+b_n+c_n $x^{-1}$ , where a₁=1, b₁=1, c₁=0, and a_n=a_n-1+b_n-1, b_n=a_n-1+b_n-1+c_n-1, c_n=a_n-1+c_n-1, for n=2,3,... .

I have got nowhere with this. I am trying to use induction. Is there a typing mistake in the question.

**emakarov** · February 22nd 2013, 08:24 AM

Originally Posted by Stuck Man

Is there a typing mistake in the question.

You can answer this if you calculate $a_2$ , $b_2$ and $c_2$ and compare $x^6$ with $a_2x+b_2+c_2$ .

I think the claim should say $x^{3n}=a_nx^2+b_nx+c_n$ where $a_1=0$ and $b_1=c_1=1$ . The recurrence equations seem correct.

**Stuck Man** · February 24th 2013, 07:22 AM

I had done exactly that. Thanks for your suggestion. I will try it.

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