https://r2.erweima.ai/mathbot/image/17270246785229xxiybxp.jpg

Let’s go through the four exercises in the image one by one.

### Exercise 33
Given the sequence \((u_n)\) defined for any natural number \(n\) by:
\[ u_n = 4n^2 + (-1)^n \]
1. **Part a**: What is the value of \((-1)^n\) depending on \(n\)?
   - If \(n\) is even, \((-1)^n = 1\).
   - If \(n\) is odd, \((-1)^n = -1\).
   
2. **Part b**: Prove that for any natural number \(n\), \(u_n = 4n^2 - 1\) when \(n \geq 1\).
   - Use the definition of \(u_n = 4n^2 + (-1)^n\).
   - For even \(n\), \((-1)^n = 1\), so \(u_n = 4n^2 + 1\).
   - For odd \(n\), \((-1)^n = -1\), so \(u_n = 4n^2 - 1\).

   Conclusion:

Sequences involving powers, cosines, and limits.

Explore how to solve sequences involving powers, cosine, and limits. Detailed step-by-step solutions include bounds for cosine, limit calculations, and inequalities in sequences. Suitable for advanced high school students (Grades 11-12).

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