# Number theory proof trouble: pesty ellipsis

1. Feb 23, 2012

### aohawthorn

1. The problem statement, all variables and given/known data

Prove that

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

2. Relevant equations

This is problem 3, section 1-1 from Andrew's "Number Theory," which I'm using for self-study. It follows the section on the "Principle of Mathematical Induction", i.e. "A statement about integers is true for all integers greater than or equal to 1 if (1) it is true for the integer 1, and (ii) whenever it is true for all the integers 1,2, ..., k, then it is true for the integer k + 1." The first two proofs both involve plugging k into an expression, adding (k + 1), and setting the sum equal to the same expression with (k + 1) plugged into it.

3. The attempt at a solution

I don't need a solution - I need to understand the question! I can't figure out what the ellipsis here is representing. In other expressions in the book (e.g. 1^3 + 2^3 + 3^3 + . . . + n^3) the ellipsis obviously represents a sequence of terms leading up to one greatest value n, but I just can't see what sort of "sequence" is represented in this problem! Any help would be deeply appreciated. Again - I'm not looking for the answer, just to understand the question. Thanks a lot!

2. Feb 23, 2012

### Deveno

$$x^n - y^n = (x - y)\left(\sum_{k=1}^n\ x^{n-k}y^{k-1}\right)$$

is a version of the formula without ellipses.

3. Feb 23, 2012

### aohawthorn

Aha! Thanks so much - I see it now.