If `a_1,a_2 ...a_n` are nth roots of unity then`1/(1-a_1) +1/(1-a_2)+1/(1-a_3)..+1/(1-a_n)` is equal to

Options

A. `(n-1)/2`
B. `n/2`
C. `(2^(n)-1)/(2)`
D. none of these

Answer

A

Explanation

It is given that `1,a_(1),a_(2),………..,a_(n-1)` are the roots of the equation
`z=1^(1/n)` or `z^(n)=1` or `z^(1)-1=0` ………….(i)
Let `alpha=1/(1-z)`i.e., `z=1-1/alpha, z ne 1`. Then, `1/(1-alpha_(1)),1/(1-alpha_(2)),…..,1/(1-a_(n-1))` are the roots of the equation
`(1-1/alpha)^(n)-1=0` [putting `z=1-1/alpha` in (i)]
`rArr (alpha-1)^(n)-alpha^(n)=0`
[Putting `z=1-1/alpha` in (i)]
`rArr (alpha-1)^(n)-alpha^(n)=0`
`rArr -.^(n)C_(1)alpha^(n-1)+^(n)C_(2)alpha^(n-2)+......+(-1)^(n)^(n)C_(n)=0`...............(i)
Thus, `1/(1-alpha_(1)),1/(1-alpha_(2)), ......,1/(1-alpha_(n-1))` are the roots of equation (ii)
`therefore 1/(1-alpha_(1))+1/(1-alpha_(2))+....+1/(1-a_(n-1))=^(n)C_(2)/-(^(n)C_(1))=(n-1)/2`