A) \[i\]
B) \[0\]
C) \[-i\]
D) \[-1\]
Correct Answer: A
Solution :
\[\underset{k=1}{\mathop{\overset{6}{\mathop{\Sigma }}\,}}\,\left( \sin \frac{2k\,\pi }{7}-i\cos \frac{2k\,\pi }{7} \right)\] \[=-i\underset{k=1}{\mathop{\overset{6}{\mathop{\Sigma }}\,}}\,\left( \cos \frac{2k\,\pi }{7}+isin\frac{2k\,\pi }{7} \right)\] \[=-i\underset{k=1}{\mathop{\overset{6}{\mathop{\Sigma }}\,}}\,\,{{a}^{k}}\left( where\,\,\alpha =\cos \frac{2k\,\pi }{7}+isin\frac{2k\,\pi }{7} \right)\] \[=-i\left( \frac{\alpha (1-{{\alpha }^{6}})}{1-\alpha } \right)=-i\left( \frac{\alpha -{{\alpha }^{7}}}{1-\alpha } \right)\] \[=-i\left( \frac{\alpha -1}{1-\alpha } \right)=i\]
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