Problem 1,v, Ch.1, Spivaks 4th Ed Calculus

linwoodc3

Hi all:

Getting back into Physics so doing self study. Having an issue with Spivak's solution to Problem 1, v.
x^n - y^n= (x-y)(x^n-1+x^n-2y+....+xy^n-2+y^n-1

Got all the way to:
x^n+x^2y^n-2-x^n-2y^2-y^n

How do i get rid of the middle section?

Attached Thumbnails

 

Mark44

Your attachment is so fuzzy it's difficult to read, and the lack of parentheses above makes it harder to understand what you have written than it needs to be.

Are you trying to show that if you multiply the two factors in your first equation, you get xⁿ - yⁿ?

If so, multiplying the factors on the right gives
xⁿ + x^{n - 1}y - x^{n - 1}y + x^{n - 2}y² - x^{n - 2}y² + ... + xy^{n - 1} - xy^{n - 1} - yⁿ. Notice that instead of multiplying all of the terms in the larger factor by x, and then repeating the process by multiplying all of the terms by -y, I have instead alternated to get partial products where the exponents add up to the same value.

The first partial product is xⁿ. The next two products come from multiplying x by x^{n -2}y, resulting in x^n-1y, and by multiplying -y by x^n-1, resulting in -x^{n - 1}y. These two terms add to zero, as do the following pairs of terms.

Doing the multiplication this way makes it clear that all but the first and last terms will drop out.