If a=e^(i*8pi/11) then find re(a+a^2+a^3+a^4+a^5) I used demoivres theorem but it is getting too lenghty
Let , so that . The roots of the equation are for . The sum of all these roots is 0. If we exclude the root z=1 then the sum of the remaining roots is –1 (and therefore so is the sum of the real parts of these roots). If you draw a picture of these 10 complex roots on the unit circle in the complex plane, you will see that they consist of and , together with their complex conjugates. It follows that the real part of must be –1/2.