1. ## Induction

If $x^{3}=x+1$, prove, by induction or otherwise, that $x^{3n}$=anx+bn+cn $x^{-1}$, where a1=1, b1=1, c1=0, and an=an-1+bn-1, bn=an-1+bn-1+cn-1, cn=an-1+cn-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.

2. ## Re: Induction

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.

3. ## Re: Induction

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