The value of $\displaystyle \sum_{k=1}^6 \bigg(\sin\frac{2\pi k}{7}-i \cos \frac{2\pi k}{7}\bigg)$ is
- -1
- 0
- -i
- i
The Correct Option is D
Solution and Explanation
$=-i\bigg \{ \displaystyle \sum_{k=1}^6 e^{\frac{i2k\pi}{7}}\bigg\}=i \{ e^{i2\pi/7}+e^{i4\pi/7}+e^{i6\pi/7}$
$\hspace30mm \, +e^{i8\pi/7}+e^{i10\pi/7}+e^{i12\pi/7} \}$
$=-i \bigg \{e^{i2\pi/7}\frac{(1-e^{i12\pi/7})}{1-e^{i2\pi/7}}\bigg \}$
$=-i \bigg \{ \frac{e^{i2\pi/7}-e^{i14\pi/7}}{1-e^{i2\pi/7}}\bigg\} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, [\because \, e^{i14\pi/7=1}]$
$=-i\bigg \{ \frac{e^{i2\pi/7-1}}{1-e^{i 2\pi/7}}\bigg\}=i$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
