How do I use finite induction to prove that
$$a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i}), \forall a,b\in \Bbb{R}\space \text{and} \space \forall n \in \Bbb{N}?$$ Ok, for $n=2$ it's fine. $a^2-b^2=(a-b)(b+a)=(a-b)(\sum_{i=0}^1a^ib^{n-1-i})$.
Right, then suppose the equality valid for n. But, how to handle with $a^{n+1}-b^{n+1}$? How to use the induction hypothesis here?

Need some help!

If this question has already been answered, please, give me the link... Sorry about that.

  • $\begingroup$ I think that using induction isn't very good idea. $\endgroup$
    – Antony
    Jan 8, 2015 at 18:12
  • 3
    $\begingroup$ @Antony It is standard. Check this. $\endgroup$
    – Git Gud
    Jan 8, 2015 at 18:13

1 Answer 1



Let $S{}={}\sum_{i=0}^{n-1}a^ib^{n-1-i}$. Compute $aS-bS$.

  • $\begingroup$ Oh! Thanks @ki3i! It seems that this proves the assertion without even using induction. Very useful. Thanks! $\endgroup$
    – Derso
    Jan 8, 2015 at 18:41
  • $\begingroup$ @AndersonFelipeViveiros, You are welcome. $\endgroup$
    – ki3i
    Jan 8, 2015 at 19:33

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