If $\displaystyle x^{3}=x+1$, prove, by induction or otherwise, that $\displaystyle x^{3n}$=a_{n}x+b_{n}+c_{n}$\displaystyle x^{-1}$, where a_{1}=1, b_{1}=1, c_{1}=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.