Let's go through each statement in the image and analyze its validity.

1. Statement: "Die Folge $(a_n)_{n \in \mathbb{N}}$ mit $a_n = 5n + 5$ ist monoton wachsend."

   • Analysis: The sequence $a_n = 5n + 5$ is an arithmetic sequence with a positive slope. Therefore, it is indeed monotonically increasing as $n$ grows.
   • Conclusion: This statement is true.

2. Statement: "Die rekursiv definierte Folge $(a_n)_{n \in \mathbb{N}}$ mit $a_1 = 5$ und $a_{n+1} = 5a_n + 5$ ist nicht monoton."

   • Analysis: Starting from $a_1 = 5$, we can see that $a_{n+1}$ grows very quickly due to the multiplication by 5. The sequence will increase rapidly, but because of the recursive definition, it will remain positive and is indeed monotonically increasing.
   • Conclusion: This statement is false because the sequence is actually monotonically increasing.

3. Statement: "Die rekursiv definierte Folge $(a_n)_{n \in \mathbb{N}}$ mit $a_1 = 2$ und $a_{n+1} = -3a_n - 2$ ist monoton."

   • Analysis: The recursive definition $a_{n+1} = -3a_n - 2$ involves alternating signs (due to the multiplication by $-3$). This sequence will not be monotonic since the terms will alternate between positive and negative values.
   • Conclusion: This statement is false.

4. Statement: "Jede monotone Folge ist beschränkt."

   • Analysis: Not all monotonic sequences are bounded. For example, an arithmetic sequence with a positive common difference is monotonically increasing but unbounded. Hence, this is not a true statement.
   • Conclusion: This statement is false.

5. Statement: "Die Folge $(a_n)_{n \in \mathbb{N}}$ mit $a_n = \frac{(-1)^n}{n}$ ist beschränkt."

   • Analysis: The sequence $a_n = \frac{(-1)^n}{n}$ oscillates between positive and negative values but decreases in magnitude as $n$ increases. The terms converge to 0, so the sequence is bounded.
   • Conclusion: This statement is true.

6. Statement: "Die Folge $(a_n)_{n \in \mathbb{N}}$ mit $a_n = (-1)^n \cdot n$ ist nach oben beschränkt."

   • Analysis: The sequence $a_n = (-1)^n \cdot n$ oscillates between positive and negative values and grows without bound in both directions (positive and negative) as $n$ increases. Therefore, it is not bounded from above.
   • Conclusion: This statement is false.

Would you like further details on any of these explanations or additional questions on related topics? Here are some related questions you might consider:

1. How can we determine if a sequence is monotonic or bounded in general?
2. What are some examples of monotonic sequences that are unbounded?
3. How does the sign change in recursive sequences affect their monotonicity?
4. What are the general conditions for a sequence to be bounded above or below?
5. Can a sequence be convergent if it is not monotonic?

Tip: For recursive sequences, calculating a few terms manually can help identify patterns and behaviors, such as whether the sequence is increasing, decreasing, or alternating.