A) \[i\]
B) zero
C) \[-i\]
D) \[-1\]
Correct Answer: A
Solution :
Given expression \[\sum\limits_{k=1}^{6}{\left( \sin \frac{2k\pi }{7}-i\cos \frac{2k\pi }{7} \right)}\] \[=-\sum\limits_{k=1}^{6}{\left( \cos \frac{2\pi k}{7}+i\sin \frac{2\pi k}{7} \right)}=-i\] \[\sum\limits_{k=1}^{6}{{{e}^{\frac{i\,2k\pi }{7}}}}\] \[=-[1+\alpha +{{\alpha }^{2}}+...{{\alpha }^{6}}-1]\] \[=-\left[ \frac{1-{{\alpha }^{7}}}{1-\alpha }-1 \right]=-i\,[0-i]=i\]
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