Step 1: Recognize the given expression.
The expression \( x_k = \cos \left( \frac{2\pi k}{n} \right) + i \sin \left( \frac{2\pi k}{n} \right) \) is the general form of a complex number on the unit circle in the complex plane, and it represents the \( n \)-th roots of unity. These roots of unity are the values that satisfy the equation:
\[
z^n = 1,
\]
where \( z \) is a complex number.
Step 2: Use the sum of the \( n \)-th roots of unity.
The sum of all the \( n \)-th roots of unity is known to be zero. This result can be derived from the geometric interpretation of these roots on the unit circle, or from the fact that they are the solutions to the equation \( z^n = 1 \), and the sum of all roots of any polynomial equation is zero (for a polynomial with no constant term). Specifically,
\[
\sum_{k=1}^{n} \left( \cos \left( \frac{2\pi k}{n} \right) + i \sin \left( \frac{2\pi k}{n} \right) \right) = 0.
\]
Step 3: Conclusion.
Thus,
\[
\sum_{k=1}^{n} x_k = 0.
\]
The correct answer is (c) 0.
A remote island has a unique social structure. Individuals are either "Truth-tellers" (who always speak the truth) or "Tricksters" (who always lie). You encounter three inhabitants: X, Y, and Z.
X says: "Y is a Trickster"
Y says: "Exactly one of us is a Truth-teller."
What can you definitively conclude about Z?
Consider the following statements followed by two conclusions.
Statements: 1. Some men are great. 2. Some men are wise.
Conclusions: 1. Men are either great or wise. 2. Some men are neither great nor wise. Choose the correct option:
"In order to be a teacher, one must graduate from college. All poets are poor. Some Mathematicians are poets. No college graduate is poor."
Which of the following is true?