x^n - 1 / x - 1?

Question

i know the answer but i want to know the proof/reason.

Answer 1

Think of a geometric progression.
How do you add up 1 + r + r^2 + .. + r^(n-1)? There are n terms, with a common ratio of r, so it is (r^n - 1)/(r-1). So what you want is precisely 1 + x + .. + x^(n-1).

You just could do a long division as well.

You could also use this 'trick' (which is used in the proof of the geometric series sum above):

x^n - 1 = (x^n + x^(n-1) + .. + x^2 + x) - (x^(n-1) + .. + x + 1), since all terms cancel out except the first x^n and the last -1.
But the first part in brackets is exactly the same as the second, except for a factor of x:
x^n - 1 = (x^(n-1) + .. + x + 1)(x-1)
So (x^n - 1)/(x-1) = x^(n-1) + .. + x + 1.

Answer 2

Look at x^3 - 1 / x-1 for the pattern. You get x² + x + 1. Look at how multiplying that by x-1 gives you terms like x² and -x² that cancel out. A formal proof would probably have to be inductive.

Answer 3

There is no "proof" to a question. There is a proof to the answer, and that is to go through the steps. If that's what you're asking for then...

(x^n)-1 / x-1... uh...

Well, if n = 2, then the answer is x+1...
.........if n = 3, then the answer is... hmm... what is the answer?

Answer 4

that is not a question :o
you must equal that expression to a number or something in order to get at least an equation...