Prove that, for integral value of `n ge1`, all the roots of the equation `nz^(n) =1 + z+ z^2 +….+z^(n)` lie within the circle `|z|=(n)/(n-1)`.

The roots of the equation z^(n)=(z+3)^(n)

Prove that none of the roots of the equation z^(n) = 2(1+ z+z^2 + …...+ z^(n-1)), n gt 1 , lies outside the circle |z|=3 .

If n>1, show that the roots of the equation z^(n)=(z+1)^(n) are collinear.

If nin Ngt1, find the sum of real parts of the roots of the equation z^(n)=(z+1)^(n).

If n is a positive integer greater than unity z is a complex number satisfying the equation z^(n)=(z+1)^(n), then

In equation (Z - 1)^(n) = Z^(n) = 1 (AA n in N) has solution, then n can be

Lt a be a complex number such that |a|<1a n dz_1, z_2z_, be the vertices of a polygon such that z_k=1+a+a^2+...+a^(k-1) for all k=1,2,3, T h e nz_1, z_2 lie within the circle (a)|z-1/(1-a)|=1/(|a-1|) (b) |z+1/(a+1)|=1/(|a+1|) (c) |z-1/(1-a)|=|a-1| (d) |z+1/(a+1)|=|a+1|

zo is one of the roots of the equation z^(n)cos theta0+z^(n-1)cos theta2+......+z cos theta(n-1)+cos theta(n)=2, where theta in R, then (A)|z0| (1)/(2)(C)|z0|=(1)/(2)