KCET · Maths · Complex Number
The value of \(\sum_{k=1}^{6}\left(\sin \frac{2 k \pi}{7}-i \cos \frac{2 k \pi}{7}\right)\) is
- A \(\mathrm{i}\)
- B 0
- C \(-\mathrm{i}\)
- D \(-1\)
Answer & Solution
Correct Answer
(A) \(\mathrm{i}\)
Step-by-step Solution
Detailed explanation
\(\sum_{k=1}^{6}\left(\sin \frac{2 k \pi}{7}-i \cos \frac{2 k \pi}{7}\right)\)
\[
=-i \sum_{k=1}^{6}\left(\cos \frac{2 k \pi}{7}+i \sin \frac{2 k \pi}{7}\right)
\]
\[
\begin{aligned}
&=-\mathrm{i} \sum_{\mathrm{k}=1}^{6} \alpha^{\mathrm{k}}\left(\text { where } \alpha=\cos \frac{2 \mathrm{k} \pi}{7}+\mathrm{i} \sin \frac{2 \mathrm{k} \pi}{7}\right) \\
&=-\mathrm{i}\left(\frac{\alpha\left(1-\alpha^{6}\right)}{1-\alpha}\right)=-\mathrm{i}\left(\frac{\alpha-\alpha^{7}}{1-\alpha}\right) \\
&=-\mathrm{i}\left(\frac{\alpha-1}{1-\alpha}\right)=\mathrm{i}
\end{aligned}
\]
\[
=-i \sum_{k=1}^{6}\left(\cos \frac{2 k \pi}{7}+i \sin \frac{2 k \pi}{7}\right)
\]
\[
\begin{aligned}
&=-\mathrm{i} \sum_{\mathrm{k}=1}^{6} \alpha^{\mathrm{k}}\left(\text { where } \alpha=\cos \frac{2 \mathrm{k} \pi}{7}+\mathrm{i} \sin \frac{2 \mathrm{k} \pi}{7}\right) \\
&=-\mathrm{i}\left(\frac{\alpha\left(1-\alpha^{6}\right)}{1-\alpha}\right)=-\mathrm{i}\left(\frac{\alpha-\alpha^{7}}{1-\alpha}\right) \\
&=-\mathrm{i}\left(\frac{\alpha-1}{1-\alpha}\right)=\mathrm{i}
\end{aligned}
\]
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