If z is one of the roots of unity with index n, find the sum

[tonit]

Homework Statement

Given the fact that z is one of the n-th roots of unity, find the sum below:
1 + 2z + 3z² + ... + nz^n-1

Homework Equations

(1-x)(1+x+...+x^n-1) = 1 - xⁿ

The Attempt at a Solution

honestly I don't know how to do this. any help is appreciated

 

[Eng1]

..the hint for the solution is called complete induction. ;)

First of all you starting to show that the beginning of the sequence is true.
After that you show that its also true for n+1...

Try to make some sort of attempt to solve it...

 

[darkxponent]

Divide eqn 2 with (1-x) and try solvin it using some calculus

 

[darkxponent]

Yes u can use induction also. But try solving it using calculus. It is simpler and more interestring

 

[tonit]

what I'm trying to solve is this
1 + 2z + 3z² + ... + nz^n-1

 

[HallsofIvy]

Yes, you said that initially and you have two different suggestions as to how to do that. Have you tried either?

 

[tonit]

I don't know how to apply induction to a sum. there is no "=" to prove. I have to find the sum, not prove something given. That's why I don't know how to apply induction.

 

[darkxponent]

Homework Equations

(1-x)(1+x+...+x^n-1) = 1 - xⁿ

There is one relevant eqn missing

 

[I like Serena]

I don't know how to apply induction to a sum. there is no "=" to prove. I have to find the sum, not prove something given. That's why I don't know how to apply induction.

I'm guessing that you're supposed to find a formula for the series (without 3 dots in it).

 

[morphism]

Hint: What do you get if you differentiate x+x^2+...+x^n?