(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

a^{-1}a = aa^{-1}= 1

Cross-multiplication rule

Cancellation law for multiplication

3. 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:

Thanks in advance for your help!

- 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]

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# Homework Help: Proof of (x^n - 1) / (x - 1) = x^n-1 + x^n-2 + ... + x + 1

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