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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
+285
+4

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
+95
+2

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

WorldEndSymphony  Aug 22, 2022