Question

How do you divide `(x^m - 1) / (x - 1)`?

Answer 1

If `m` is a positive integer, then:

`(x^m-1)/(x-1) = x^(m-1)+x^(m-2)+...+x+1 = sum_(k=0)^(m-1) x^k`

Explanation:

Notice that if `m` is a positive integer, then:

`(x-1) sum_(k=0)^(m-1)x^k`

`= x sum_(k=0)^(m-1)x^k - sum_(k=0)^(m-1)x^k`

`= sum_(k=1)^m x^k - sum_(k=0)^(m-1) x^k`

`= x^m + color(red)(cancel(color(black)(sum_(k=1)^(m-1)x^k))) - color(red)(cancel(color(black)(sum_(k=1)^(m-1) x^k))) - 1`

`= x^m - 1`

Dividing both ends by `(x-1)` we find:

`sum_(k=0)^(m-1)x^k = (x^m-1)/(x-1)`