- #1

jjangub

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## Homework Statement

Let w = e^((2pi*i)/n). Show that 1+2w+3w^2+...+nw^(n-1) = n/(w-1)

## Homework Equations

1+x+x^2+x^3+...+x^m = (1-x^(m+1))/(1-x) --> A

1+x+x^2+x^3+... = 1/(1-x) --> B

## The Attempt at a Solution

First of all, multiply (w-1) on both sides, then we get

w+2w^2+3w^3+...+(nw^n)-1-2w-3w^3-...-nw^(n-1) = n we simplify left side,

-1-w-w^2-w^3-...-w^n+(nw^n) = n add -1 on both sides

1+w+w^2+w^3+...+w^n-(nw^n) = -n

1+w+w^2+w^3+...+w^n = (nw^n)-n for the left side, we know from A that

(1-w^(n+1))/(1-w) = (nw^n)-n

But I can't get left side and right side equal.

Did I use the right method? Which part is wrong?

Please tell me...

Thank you