Standard Form/Expansion of x^n - y^n

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Discussion Overview

The discussion revolves around the expansion of the expression x^n - y^n, exploring potential forms similar to the known expansion for x^a - 1. Participants are examining mathematical representations and simplifications related to this expression.

Discussion Character

  • Mathematical reasoning, Technical explanation, Debate/contested

Main Points Raised

  • One participant queries whether there is a similar expansion for x^n - y^n as there is for x^a - 1.
  • Another participant suggests considering the expression when divided by y^n.
  • A participant notes that the expression can be rewritten as (\frac{x}{y})^n - 1, but expresses a desire for a more aesthetically pleasing form.
  • Another participant challenges the previous assertion, suggesting that the simplification is nearly equivalent to the original expression and hints at a possible oversight in not including y^n.
  • One participant concludes that any desired neat sum will ultimately be a rearrangement of the previously proposed form.
  • A later reply acknowledges a misunderstanding and expresses gratitude for the clarification provided by others.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement and disagreement, with some proposing alternative forms and others challenging those suggestions. The discussion remains unresolved regarding the most desirable or simplified form of the expansion.

Contextual Notes

The discussion does not resolve the assumptions or definitions regarding the expansions, nor does it clarify the conditions under which the proposed forms hold true.

Gib Z
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Just as [tex]x^a-1 = (x-1)\sum_{n=0}^{a-1} x^n[/tex], is there a similar expansion for x^n - y^n?
 
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What happens if you divide by yn?
 
[tex](\frac{x}{y})^n -1[/tex] which fits the previous form, but I was hoping i'd get something a bit nicer looking >.<
 
What's wrong with it? Once you're done simplifying, it's almost the same expression. Maybe you didn't multiply the y^n back in?

I get (x - y) sum_i x^i y^(n-1-i) [/color].
 
Last edited:
Gib Z... anyway you look at it, if you want a nice looking sum, it will only be a rearrangement of what Hurkyl proposed.
 
Ok i see it now, my bad lol. Thanks guys
 

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