Vtyjkyuk

It wonder with a finite series

$$y = x^0 + x^1 + x^2 + ... x^n-1$$

can be formulate into $$fracx^n - 1x - 1$$

But I don't understand why $x^n - 1$ could be divide by $x-1$ and always be an integer. What is the relation between $x^a - 1$ and $x - 1$. It seem like a mystery

Are there any proof, if possible a visual proof, that would make it easy to understand this relation?

For a visual proof, try thinking in the other direction.

$(x-1)cdot (x^0+x^1+x^2+dots+x^n-1)$

$ = xcdot (x^0+x^1+x^2+dots+x^n-1)-1cdot (x^0+x^1+x^2+dots+x^n-1)$

$= (colorbluex^1+colorgreenx^2+colorredx^3+dots+colororangex^n-1+x^n) - (x^0+colorbluex^1+colorgreenx^2+dots+colororangex^n-1)$

$=-x^0 + (x^1-x^1)+(x^2-x^2)+(x^3-x^3)+dots+(x^n-1-x^n-1)+x^n$

$=x^n-1$

Maybe the algebraically most obvious way to see it is as follows:

$$begineqnarray y & = & x^0 + colorbluex^1 + x^2 + ... + x^n-1 \
xcdot y & = & colorbluex^1 + x^2 + ... + x^n-1 + x^nendeqnarray$$
By subtracting the equations you get
$$ Rightarrow xy-y = (x-1)y = x^n-1$$

HINT

Let consider

$$(x-1)(x^n-1+x^n-2+ldots+x+1)$$