The value of `z` satisfying the equation `logz+logz^2+dot+logz^n=0i s`

Options

A. `cos.(4mpi)/(n(n+1)) +i sin.(4mpi)/(n(n+1)),m = 0,1,2,...`
B. `cos.(4mpi)/(n(n+1)) -i sin.(4mpi)/(n(n+1)),m = 0,1,2,...`
C. `sin.(4mpi)/(n) +i cos.(4mpi)/(n),m = 0,1,2,...`
D. 0

Answer

A

Explanation

We have,
` log z + log z^(2) + log z^(3) + ……+ log z^(n) =0`
`rArr log (z z^(2) z^(3) …..z^(n))=0`
`rArr log(z^((n(n+1))/(2))) =0`
`or z^((n(n+1))/(2)) = 1`
`= (cos 0^(@) + isin 0^(@))^((2)/(n(n+1)))`
` = (cos 2 mpi + isin 2mpi)^((2)/(n(n+1))), m = 0,1,2,3,.....`
`= cos.(4mpi)/(n(n+1)) +isin .(4mpi)/(n(n+1)), m = 0,1,2,`