Question

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

• A. -1
• B. \N
• C. -i
• D. i

Answer 1

The correct option is D i

${\sum }_{k=1}^6(sin\frac{2k\pi }{7}-icos\frac{2k\pi }{7})=-i{\sum }_{k=1}^6(cos\frac{2k\pi }{7}+isin\frac{2k\pi }{7})$
$=-i\left\{{\sum }_{k=1}^6{e}^{\frac{i2k\pi }{7}}\right\}=-i\{e^{\frac{i2k\pi }{7}}+e^{\frac{i4k\pi }{7}}+e^{\frac{i6k\pi }{7}}+e^{\frac{i8k\pi }{7}}+e^{\frac{i10k\pi }{7}}+e^{\frac{i12k\pi }{7}}\}$
$=-i\left\{e^{\frac{i2k\pi }{7}}\frac{(1-e^{\frac{i12\pi }{7}})}{1-e^{\frac{i2\pi }{7}}}\right\}$
$=-i\left\{\frac{e^{\frac{i2\pi }{7}}-e^{\frac{i14\pi }{7}}}{1-e^{\frac{i2\pi }{7}}}\right\}[\because e^{\frac{i14\pi }{7}}=1]=-i\left\{\frac{e^{\frac{i2\pi }{7}}-1}{1-e^{\frac{i2\pi }{7}}}\right\}=i$