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If \(\omega^{1997}=1\) and \(\omega \neq 1\), then evaluate \(\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} + \dots + \frac{1}{1 + \omega^{1997}}\).

 Aug 19, 2022
 #1
avatar+251 
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Roots of Unity! One of my favorite topics! We can use root transformataion for this problem!

 

w^{1997} - 1 = 0 

 

In order to achieve a polynomial with w + 1 as the roots, we plug in w - 1.

 

(w-1)^{1997} - 1 = 0

 

 

If we want a polynomial with the reciprocal of the roots, we have to flip the coefficients. 

 

(w-1)^{1997} - 1 = w^{1997} - 1997w^{1996}+....+1997w -1 - 1 = 0. 

 

Flip the coefficients of the right hand side gives us:

 

-2w^{1997} + 1997w^{1996}+.... =  0

 

Can you solve the sum of the roots of the equatoin above? :)

 Aug 21, 2022
 #2
avatar+84 
+2

thank you so much!!! for anyone interested, the answer is \(\frac{1997}{2}\)!!

WorldEndSymphony  Aug 22, 2022

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