Question

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

• A. -1
• B. 0
• C. -i
• D. i

Answer 1

Answer: (D)

$\sum _{k=1}^6(\sin \frac{2\pi k}{7}-i\cos \frac{2\pi k}{7})=\sum _{k=1}^6-i(\cos \frac{2\pi k}{7}+i\sin \frac{2\pi k}{7})$
$=-i\left\{\sum _{k=1}^6{e}^{\frac{i2k\pi }{7}}\right\}=i\{e^{i2\pi /7}+e^{i4\pi /7}+e^{i6\pi /7}$
$+e^{i8\pi /7}+e^{i10\pi /7}+e^{i12\pi /7}\}$
$=-i\left\{e^{i2\pi /7}\frac{(1-e^{i12\pi /7})}{1-e^{i2\pi /7}}\right\}$
$=-i\left\{\frac{e^{i2\pi /7}-e^{i14\pi /7}}{1-e^{i2\pi /7}}\right\}[\because e^{i14\pi /7=1}]$
$=-i\left\{\frac{e^{i2\pi /7-1}}{1-e^{i2\pi /7}}\right\}=i$