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Thread: Induction

  1. #1
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    Induction

    If $\displaystyle x^{3}=x+1$, prove, by induction or otherwise, that $\displaystyle x^{3n}$=anx+bn+cn$\displaystyle x^{-1}$, 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.
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  2. #2
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    Re: Induction

    Quote Originally Posted by Stuck Man View Post
    Is there a typing mistake in the question.
    You can answer this if you calculate $\displaystyle a_2$, $\displaystyle b_2$ and $\displaystyle c_2$ and compare $\displaystyle x^6$ with $\displaystyle a_2x+b_2+c_2$.

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

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