If z+1/z=1, find z^3, and then z^1000+1/z^1000?

2 Answers
Aug 9, 2018

Please see the explanation below.

Explanation:

Let

z=(costheta+isintheta)

1/z=(costheta-isintheta)

z+1/z=2costheta=1

costheta=1/2

theta=pi/3, [mod 2pi]

Therefore,

z=cos(pi/3)+isin(pi/3)

And

z^3=cos(3*pi/3)+isin(3*pi/3)=-1

Avcoording to Demoivre' s Theorem

z^1000=cos(1000*pi/3)+isin(1000*pi/3)

1/z^1000=cos(1000*pi/3)-isin(1000*pi/3)

z^1000+1/z^1000=2cos(1000/3pi)=-2*0.5=-1

Aug 9, 2018

z^3 = -1" " and " "z^1000 + 1/z^1000 = -1

Explanation:

Given:

z+1/z = 1

Note that:

z^3+1 = (z+1)(z^2-z+1) = (z+1)z(z+1/z-1) = 0

So:

z^3 = -1

So:

z^1000+1/z^1000 = z^(3*333+1)+1/z^(3*333+1)

color(white)(z^1000+1/z^1000) = (-1)^333(z+1/z)

color(white)(z^1000+1/z^1000) = -(z+1/z)

color(white)(z^1000+1/z^1000) = -1