- #1

## Homework Statement

The problem comes from Lang's Basic Mathematics, chapter 1, paragraph 6 (multiplicative inverses) and simply asks to prove the relation:

(x

^{n}- 1) / (x - 1) = x

^{n - 1}+ x

^{n - 2}+ ... + x + 1

## Homework Equations

a

^{-1}a = aa

^{-1}= 1

Cross-multiplication rule

Cancellation law for multiplication

## The Attempt at a Solution

The solution is actually given at the back of the book, but there's a couple of simplifications I have trouble understanding:

(x - 1) (x

^{n-1}+ x

^{n-2}+ ... + x + 1)

= x(x

^{n-1}+ x

^{n-2}+ ... + x + 1) - (x

^{n-1}+ x

^{n-2}+ ... + x + 1)

= x

^{n}+ x

^{n-1}+ ... + x - x

^{n-1}- x

^{n-2}- ... - x - 1

= x

^{n}- 1

The two things I don't understand are:

- Shouldn't the result of x(x
^{n-1}+ x^{n-2}+ ... + x + 1) include x^{2}? [line 2] - How is x
^{n-2}excluded from the final result? [line 3]