+99 If \(\omega^{1997}=1\) and \(\omega \neq 1\), then evaluate \(\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} + \dots + \frac{1}{1 + \omega^{1997}}\).
+288 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? :)
+99 thank you so much!!! for anyone interested, the answer is \(\frac{1997}{2}\)!!
