- #1
Stefk
Homework Statement
The problem comes from Lang's Basic Mathematics, chapter 1, paragraph 6 (multiplicative inverses) and simply asks to prove the relation:
(xn - 1) / (x - 1) = xn - 1 + xn - 2 + ... + x + 1
Homework Equations
a-1a = 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) (xn-1 + xn-2 + ... + x + 1)
= x(xn-1 + xn-2 + ... + x + 1) - (xn-1 + xn-2 + ... + x + 1)
= xn + xn-1 + ... + x - xn-1 - xn-2 - ... - x - 1
= xn - 1
The two things I don't understand are:
- Shouldn't the result of x(xn-1 + xn-2 + ... + x + 1) include x2? [line 2]
- How is xn-2 excluded from the final result? [line 3]