Question

Let \( a_n \) be the sequence given by \[ a_n = \max \left( \sin \left( \frac{n \pi}{3} \right), \cos \left( \frac{n \pi}{3} \right) \right), \quad n \geq 1. \] Then which of the following statements is/are TRUE about the subsequences \( \{a_{6n-1}\} \) and \( \{a_{6n+1}\} \)?

• A. Both the subsequences are convergent
• B. Only one of the subsequences is convergent
• C. \( \{a_{6n-1}\} \) converges to \( -\frac{1}{2} \) and \( \{a_{6n+1}\} \) converges to \( \frac{1}{2} \)
• D. \( \{a_{6n+4}\} \) converges to \( \frac{1}{2} \)

Hint:

When analyzing periodic sequences, break them into subsequences based on the period and check their limits.

Answer 1

The Correct Option is A

Step 1: Analyze the sequence \( a_n \).
The sequence \( a_n = \max \left( \sin \left( \frac{n \pi}{3} \right), \cos \left( \frac{n \pi}{3} \right) \right) \) repeats periodically with a period of 6. The values of \( \sin \left( \frac{n \pi}{3} \right) \) and \( \cos \left( \frac{n \pi}{3} \right) \) repeat every 6 terms.
Step 2: Evaluate the subsequences.
- For \( a_{6n-1} \), the values of \( \sin \left( \frac{(6n-1) \pi}{3} \right) \) and \( \cos \left( \frac{(6n-1) \pi}{3} \right) \) converge to \( -\frac{1}{2} \). - For \( a_{6n+1} \), the values of \( \sin \left( \frac{(6n+1) \pi}{3} \right) \) and \( \cos \left( \frac{(6n+1) \pi}{3} \right) \) converge to \( \frac{1}{2} \).
Step 3: Conclusion.
Thus, the correct answer is \( \boxed{(C)} \).