Proof for x^n-y^n=(x-y)(x^n-1+ +y^n-1)

  • Thread starter moviefan91
  • Start date
  • #1
Proof for x^n-y^n=(x-y)(x^n-1+....+y^n-1)

Homework Statement


The question asks to prove that for any n[itex]\geq[/itex]1,
[itex]x^{n}[/itex]-[itex]y^{n}[/itex]=(x-y)([itex]x^{n-1}[/itex]+[itex]x^{n-2}[/itex]y+...+[itex]y^{n-1}[/itex])


Homework Equations


[itex]x^{n}[/itex]-[itex]y^{n}[/itex]=(x-y)([itex]x^{n-1}[/itex]+[itex]x^{n-2}[/itex]y+...+[itex]y^{n-1}[/itex])


The Attempt at a Solution



So far, I used induction.
So for n=1, x-y=x-y

Second step, I assume that n=k is true:
[itex]x^{k}[/itex]-[itex]y^{k}[/itex]=(x-y)([itex]x^{k-1}[/itex]+[itex]x^{k-2}[/itex]y+...+[itex]y^{k-1}[/itex])

I get stuck at n=k+1.

[itex]x^{k+1}[/itex]-[itex]y^{k+1}[/itex]=(x-y)([itex]x^{k}[/itex]+[itex]x^{k-1}[/itex]y+...+[itex]y^{k}[/itex])

When I expand RHS, I get:
[itex]x^{k+1}[/itex]-[itex]x^{k}[/itex]y+[itex]x^k{}[/itex]y-[itex]x^{k-1}[/itex][itex]y^{2}[/itex]+...+x[itex]y^{k}[/itex]-[itex]y^{k+1}[/itex]

I think that I need to cancel things so I can be left only with [itex]x^{k+1}[/itex]-[itex]y^{k+1}[/itex], but I always have terms in the middle which do not cancel out.

What am I doing wrong?
 

Answers and Replies

  • #2
lanedance
Homework Helper
3,304
2


you need to work from the the n case to the n+1 (or vice versa)

i haven't worked it, but how about noticing:
[tex](x+y)(x^{k} -y^{k}) = x^{k+1} -xy^{k} -x^{k}y -y^{k+1}[/tex]

then you have
[tex]x^{k+1} -y^{k+1} = (x+y)(x^{k} -y^{k})-x^{k}y +xy^{k}[/tex]

then see if you can work it into the required form...
 

Related Threads on Proof for x^n-y^n=(x-y)(x^n-1+ +y^n-1)

  • Last Post
Replies
16
Views
8K
  • Last Post
Replies
6
Views
2K
  • Last Post
2
Replies
28
Views
9K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
6
Views
2K
Replies
5
Views
2K
Replies
13
Views
2K
Replies
5
Views
2K
  • Last Post
Replies
10
Views
1K
Top