I'm banging my head against the wall on this one.
43. if z = exp(2*Pi*i/5), then 1 + z + z^2 + z^3 + 5z^4 + 4z^5 + 4z^6 + 4z^7 + 4z^8 + 5z^9 = ...
The answer is -5*exp(3*Pi*i/5), and I can't seem to get there.
I haven't had complex analysis, so I only know what I've picked up from using it in other classes. I can get the given expression to equal:
5(1 + exp(Pi*i/5) + exp(2*Pi*i/5) + 2*exp(3*Pi*i/5) + exp(4*Pi*i/5))
but I have no idea how to combine those exponentials. I know they're all 5th roots of unity, but I don't see how that helps. Any ideas?
(Also, sorry, I've never used LaTex, so this looks crappy. I'll learn it when I need it.)