Multiple choice

Let ${ z }{ k }\left( k=0,1,2,...,6 \right) $ be the roots of the equation ${ \left( z+1 \right) }^{ 7 }+{ z }^{ 7 }=0$, then $\displaystyle \sum _{ k=0 }^{ 6 }{ Re\left( { z }{ k } \right) } $ is equal to

  1. $0$
  2. $\displaystyle \frac { 3 }{ 2 } $
  3. $\displaystyle -\frac { 7 }{ 2 } $
  4. $\displaystyle \frac { 7 }{ 2 } $
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C Correct answer
Explanation

The roots of (z+1)^7 + z^7 = 0 satisfy |z+1| = |z|. This implies Re(z) = -1/2. Since there are 7 roots, the sum of the real parts is 7 * (-1/2) = -7/2.

AI explanation

Expanding (z+1)^7 + z^7 = 0 yields z^7 + 7z^6 + 21z^5 + 35z^4 + 35z^3 + 21z^2 + 7z + 1 + z^7 = 0, which simplifies to 2z^7 + 7z^6 + 21z^5 + 35z^4 + 35z^3 + 21z^2 + 7z + 1 = 0. Using Vieta's formulas, the sum of the roots z0 through z6 is given by the ratio of the coefficient of z^6 to the coefficient of z^7, which is -7/2. The real part of a sum is the sum of the real parts, meaning the required sum of the real parts of all seven roots is -7/2.

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