The solutions of zn 1 are known as nth roots of unity For a

The solutions of z^n = 1 are known as n-th roots of unity. For any given integer n, show the sum of the n-th roots of unity is zero.

Solution

An (n)th root of utility is a complex no satisfied the eqn

Z^(n) - 1 = 0

Z^(n) = 0

By fundamental theourem of algebra there are (n)th roots of utility

since 1 = e^(2*pi*i)

write (n)th root w = e^{(2*pi*i)/n}

All other (n)th root w are given by the intiger power of this root b/w (0) and (n-1)

the root are 1,w,w^(2).......w^(n-1)

y factorise the eqn Z^(n) -1 = 0

Z^(n) -1 = (z-1)(z^(n-1)+(z^(n-2)........+ z+1) = 0

This means that Ball the (n) th roots of utility other than (1) satisfied the eqn .

1+w+w^(2)+......w^(n-1) = 0

Hence proved