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If $z_r(r=1,2,......7)$ are the roots of $(z_r+1)^7+z_r^7=0$, then $\sum\limits_{r=1}^{7} Re(z_r)=?$

$\begin{array}{1 1}(A)0 \\ (B) \large\frac{3}{2} \\ (C) \large\frac{7}{2} \\(D) \large\frac{-7}{2} \end{array}$

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Given: $(z_r+1)^7+z_r^7=0$
$z_r=x_r+iy_r$
$\Rightarrow\:(z_r+1)^7=-z_r^7$
$\Rightarrow\:|(z_r+1)|^7=|z_r|^7$
$\Rightarrow\:|(z_r+1)|=|z_r|$
$\Rightarrow\:|(z_r+1)|^2=|z_r|^2$
$\Rightarrow\:(x_r+1)^2+y_r^2=x_r^2+y_r^2$
$\Rightarrow\:2x_r+1=0$
$\Rightarrow\:x_r=\large-\frac{1}{2}$$\forall r=1,2,......7$
$\sum\limits _{r=1}^{7} R_e(z_r)=\sum\limits_{r=1}^{7}x_r=-\large\frac{7}{2}$
answered Aug 6, 2013 by rvidyagovindarajan_1
 

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