Question

The value of $\displaystyle \sum_{k=1}^6 \bigg(\sin\frac{2\pi k}{7}-i \cos \frac{2\pi k}{7}\bigg)$ is

• A. -1
• B. 0
• C. -i
• D. i

Answer 1

Answer: (D)

i

Answer 2

$\displaystyle \sum_{k=1}^6\bigg(sin\frac{2\pi k}{7}-i cos \frac{2\pi k}{7}\bigg)=\displaystyle \sum_{k=1}^6 -i \bigg(cos\frac{2\pi k}{7}+i sin \frac{2\pi k}{7}\bigg)$
=-i\bigg \\{ \displaystyle \sum_{k=1}^6 e^{\frac{i2k\pi}{7}}\bigg\\}=i \\{ e^{i2\pi/7}+e^{i4\pi/7}+e^{i6\pi/7}
\hspace30mm \, +e^{i8\pi/7}+e^{i10\pi/7}+e^{i12\pi/7} \\}
=-i \bigg \\{e^{i2\pi/7}\frac{(1-e^{i12\pi/7})}{1-e^{i2\pi/7}}\bigg \\}
=-i \bigg \\{ \frac{e^{i2\pi/7}-e^{i14\pi/7}}{1-e^{i2\pi/7}}\bigg\\} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, [\because \, e^{i14\pi/7=1}]
=-i\bigg \\{ \frac{e^{i2\pi/7-1}}{1-e^{i 2\pi/7}}\bigg\\}=i