# Spivak, problem 1v

1. Dec 19, 2007

### rocomath

Prove the following:

$$x^n - y^n = (x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})$$

Hmm ... I factored x then -y and came out with:

$$x^{n}+...x^{2}y^{n-2}-x^{n-2}y^{2}...-y^{n}$$

Argh. What's up with the middle part? I'm not sure where to go from here.

Last edited: Dec 19, 2007
2. Dec 19, 2007

### mathboy

Direct expansion will verify it.

Last edited: Dec 19, 2007
3. Dec 19, 2007

### rocomath

Oops, I mistyped something.

4. Dec 19, 2007

### chickendude

Putting the right hand side in sigma form

$$(x-y) \sum_{i=1}^{n} x^{n-i}y^{i-1}$$

$$\sum_{i=1}^{n} x^{n-i+1}y^{i-1} - x^{n-i}y^{i}$$

$$\sum_{i=0}^{n-1} x^{n-i}y^{i} - \sum_{i=1}^{n}x^{n-i}y^{i}$$

$$x^n - y^n + \sum_{i=1}^{n-1} x^{n-i}y^{i} - \sum_{i=1}^{n-1}x^{n-i}y^{i}$$

$$x^n - y^n$$

5. Dec 19, 2007

### rocomath

Too bad I have no idea what that means :-X I guess I'll just have to wait till I get to those types of methods. Thanks tho.

6. Dec 19, 2007

### morphism

Just expand the right side as mathboy suggested. chickendude just did it in abbreviated form.

7. Dec 19, 2007

### rocomath

What does it mean to "expand" it. I think that's in the next chapter, so I'll just go back to it.

8. Dec 19, 2007

### morphism

It means to multiply the thing out, e.g. (x-y)(x+y) = x^2 + xy - yx - y^2. Have you done basic algebra? If not, I don't think Spivak is right for you just yet.

9. Dec 19, 2007

### rocomath

Uh. That is exactly what I did smart ***. And what I got in the middle makes no sense to me.

10. Dec 19, 2007

### morphism

No reason to get snappy. :tongue2: Everything between x^n and -y^n will cancel off, e.g. we're going to get x^(n-1) y and -y x^(n-1), etc. Try it out for n=3 to get a feel for it.

11. Feb 4, 2008

### PFStudent

Hey,

Sorry, for posting a little late. This problem is interesting out of what textbook did you get this problem from?

Thanks,

-PFStudent

12. Feb 4, 2008

### Defennder

It's Spivak's Calculus. He said so in the title.

13. Feb 4, 2008

### PFStudent

Hey,

Ahem. Spivak (that is, Michael Spivak) is the author of several calculus titles,

Calculus
Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus
A Comprehensive Introduction to Differential Geometry, Volumes 1-4: 3rd Edition
The Hitchhiker's Guide to Calculus
Calculus: Calculus of Infinitesimals

And no, he did not mention it was specifically from the text, Calculus.

What I wanted to know was which one of his texts had the problem. It was already obvious that it was from one of his several calculus texts, however which one was not.

Thanks,

-PFStudent

Last edited: Feb 4, 2008