Equation `x^(n)-1=0, ngt1, n in N," has roots "1,a_(1),a_(2),…,a_(n-1).`
The value of `overset(n-1)underset(r=1)sum(1)/(2-a_(r))` is

Options

A. `(n)/(4)`
B. `(n(n-1))/(2)`
C. `(n-1)/(2)`
D. none of these

Explanation

`"From "(2),(nx^(n-1))/(x^(n)-1)-(1)/(x-1)=(1)/(x-a_(1))+(1)/(x-a_(2))+...+(1)/(x-a_(n-1))`
`"or "(nx^(n-1)-1(1+x+x^(2)+...+x^(n-1)))/(x^(n)-1)`
`=(1)/(x-a_(1))+(1)/(x-a_(2))+...+(1)/(x-a_(n-1))`
`"or "underset(xrarr1)lim(nx^(n-1)-1(1+x+x^(2)+...+x^(n-1)))/(x^(n)-1)`
`=underset(xrarr1)lim((1)/(x-a_(1))+(1)/(x-a_(2))+...+(1)/(x-a_(n-1)))`
`"or "underset(xrarr1)lim(n(n-1)x^(n-2)-1(1+2x+...+(n-1)x^(n-2)))/(nx^(n-1))`
`=(1)/(1-a_(1))+(1)/(1-a_(2))+...+(1)/(1-a_(n-1))`
(applying L Hopital's rule on L.H.S.)
`"or "(n(n+1)-(1+2+...+(n-1)))/(n)`
`=(1)/(1-a_(1))+(1)/(1-a_(2))+...+(1)/(1-a_(n-1))`
`"or "(1)/(1-a_(1))+(1)/(1-a_(2))+...+(1)/(1-a_(n-1))=(n-1)/(2)`