Question

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

• A. $-1$
• B. $0$
• C. $-i$
• D. $i$

Answer 1

The correct option is D

$i$

Explanation for the correct answer:

Step 1 : Simplify the given summation

Let $S=$$\sum _{k=1}^6\left(\sin \left(\frac{2\mathrm{\pi k}}{7}\right)-i\cos \left(\frac{2\mathrm{\pi k}}{7}\right)\right)$

$\Rightarrow$ $S=$$\underset{k=1}{\overset{6}{\sum \left(-i\right)}}\left(\cos \left(\frac{2\mathrm{\pi k}}{7}\right)+i\sin \left(\frac{2\mathrm{\pi k}}{7}\right)\right)$

$\Rightarrow$ $S=$$-i\sum _{k=1}^6{e}^{\frac{i2\mathrm{\pi k}}{7}}$ $\left[\because e^{i\theta }=\cos \theta +i\sin \theta \right]$

$\Rightarrow$ $S=$$-i\left(e^{i\frac{2\mathrm{\pi }}{7}}+e^{i\frac{4\mathrm{\pi }}{7}}+e^{i\frac{6\mathrm{\pi }}{7}}+e^{i\frac{8\mathrm{\pi }}{7}}+e^{i\frac{10\mathrm{\pi }}{7}}+e^{i\frac{12\mathrm{\pi }}{7}}\right)$

Step 2: Use the formula for sum of finite terms in a geometric progression

The above sequence is a geometric progression with the first term $a=e^{i\frac{2\mathrm{\pi }}{7}}$ and common ratio $r=e^{i\frac{2\mathrm{\pi }}{7}}$, $n=6$.

The sum of the geometric progression is given as

$S_{gp}=a\frac{\left(1-r^n\right)}{\left(1-r\right)}$

Substituting the values we get

$S_{gp}=e^{i\frac{2\mathrm{\pi }}{7}}\frac{\left(1-e^{i\frac{12\mathrm{\pi }}{7}}\right)}{\left(1-e^{i\frac{2\mathrm{\pi }}{7}}\right)}$

$\Rightarrow$ $S=$$-i\left(\frac{e^{i{\displaystyle \frac{2\mathrm{\pi }}{7}}}-e^{i{\displaystyle \frac{14\mathrm{\pi }}{7}}}}{\left(1-e^{i\frac{2\mathrm{\pi }}{7}}\right)}\right)$

Step 3: Use Euler's representation of complex numbers

$e^{i\frac{14\mathrm{\pi }}{7}}=e^{i2\mathrm{\pi }}=\cos 2\mathrm{\pi }+\mathrm{isin}2\mathrm{\pi }=1$

$\Rightarrow$ $S=$$-i\left(\frac{e^{i{\displaystyle \frac{2\mathrm{\pi }}{7}}}-1}{1-e^{i\frac{2\mathrm{\pi }}{7}}}\right)$

$\Rightarrow$ $S=$$-i\left(-1\right)$

$\Rightarrow$ $S=$$i$

Hence the value of the summation $\sum _{k=1}^6\left(\sin \left(\frac{2\mathrm{\pi k}}{7}\right)-i\cos \left(\frac{2\mathrm{\pi k}}{7}\right)\right)$ is $i$.

Hence, option $\left(D\right)$ is the correct answer.