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Topic: complex numbers
Replies: 8   Last Post: Jan 31, 2007 7:33 PM

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Michael Olea

Posts: 57
Registered: 12/13/04
Re: complex numbers
Posted: Jan 30, 2007 9:45 PM
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Michael Olea wrote:

> jennifer wrote:
>

>> How would i evaluate the following:
>>
>> 1+2w+3w^2+...+nw^n-1, where w is an nth root of unity.
>>
>> apparently, the solution is: n(n+1)/2 if w= 1
>>
>> I don't know how they got this? Any hints please?

>
> I can give you a hint for this part. When w=1 the sum is:
>
> 1+2+3+...+n
>
> One hint is to notice the pattern:
>
> 1+n = n+1
> 2+(n-1) = n+1
> 3+(n-2)= n+1
> ...
>
> And treat the cases n is odd and n is even separately.


Hm. I have, up till now, always had to stop and think a second or two about
the boundary conditions for n odd or even, but it occurs to me there is a
simpler way to see the result. Line up the numbers like this (monospaced
font recommended):

1 2 3 ... (n-2) (n-1) n
n (n-1) (n-2) ... 3 2 1
==============================

Add the top row to the bottom row, which gives n summands all adding up to
(n+1). That will be twice the value of the desired result...

-- Michael




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