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The question presented provides two sequences \( a_n \) and \( b_n \), defined as follows:

\[
a_n = \frac{2n - 1}{4n} + \frac{n - 1}{4n}
\]
\[
b_n = 2n \cdot a_n
\]

You are asked to determine whether the following six statements are true or false. Let's analyze the sequences step by step:

### Step 1: Simplification of \( a_n \)
First, simplify \( a_n \):

\[
a_n = \frac{2n - 1}{4n} + \frac{n - 1}{4n} = \frac{2n - 1 + n - 1}{4n} = \frac{3n - 2}{4n}
\]

Thus,
\[
a_n = \frac{3n - 2}{4n} = \frac{3}{4} - \frac{1}{2n}
\]

### Step 2: Simplification of \( b_n \)
Now simplify \( b_n = 2n \cdot a_n \):

\[
b_n = 2n \cdot \left( \frac{3}{4} - \frac{1}{2n} \right) = \frac{3n}{2} - 1
\]

Thus, we have the simplified forms:
\[
a_n = \frac{3}{4} - \frac{1}{2n}, \quad b_n = \frac{3n}{2} - 1
\]

### Analysis of the Statements:

1. **Convergence of \( a_n \):**  
   As \( n \to \infty \), the term \( \frac{1}{2n} \to 0 \), so \( a_n \to \frac{3}{4} \). Hence, the sequence \( (a_n)_{n=1}^{\infty} \) converges.  
   **True**.

2. **Boundedness of \( |a_n - \frac{3}{4}| \) for \( n \geq 30 \):**  
   From the simplified form of \( a_n \), we have \( a_n = \frac{3}{4} - \frac{1}{2n} \). The absolute difference \( |a_n - \frac{3}{4}| = \frac{1}{2n} \), and for \( n \geq 30 \),  
   \[
   \frac{1}{2n} \leq \frac{1}{60} < \frac{1}{50}
   \]
   So the inequality holds.  
   **True**.

3. **Monotonicity of \( a_n \):**  
   \( a_n = \frac{3}{4} - \frac{1}{2n} \) is a decreasing function of \( n \) because \( \frac{1}{2n} \) decreases as \( n \) increases. Hence, \( (a_n)_{n=1}^{\infty} \) is not monotonically increasing.  
   **False**.

4. **Boundedness of \( b_n \):**  
   \( b_n = \frac{3n}{2} - 1 \), which grows without bound as \( n \to \infty \). Thus, \( (b_n)_{n=1}^{\infty} \) is not bounded.  
   **False**.

5. **Arithmetic progression of \( b_n \):**  
   The general form of an arithmetic sequence is \( b_n = a + (n-1)d \). From the expression for \( b_n = \frac{3n}{2} - 1 \), it can be rewritten as:  
   \[
   b_n = \frac{3}{2}n - 1
   \]
   which is an arithmetic sequence with a common difference of \( \frac{3}{2} \).  
   **True**.

6. **Limit of \( \frac{a_n}{n} \):**  
   Since \( a_n = \frac{3}{4} - \frac{1}{2n} \), we analyze the limit:  
   \[
   \lim_{n \to \infty} \frac{a_n}{n} = \lim_{n \to \infty} \frac{\frac{3}{4} - \frac{1}{2n}}{n} = \lim_{n \to \infty} \left( \frac{3}{4n} - \frac{1}{2n^2} \right) = 0
   \]
   Hence, the limit is zero.  
   **True**.

### Summary of the Truth of the Statements:
1. True  
2. True  
3. False  
4. False  
5. True  
6. True

Would you like more details on any step or clarification?

Here are 5 questions related to this analysis:
1. How do you determine if a sequence converges?
2. What is the geometric meaning of an arithmetic progression?
3. Why is the sequence \( b_n \) unbounded?
4. What is the significance of the limit \( \lim_{n \to \infty} \frac{a_n}{n} \)?
5. Can the sequence \( a_n \) be increasing for any range of \( n \)?

**Tip**: Always check both the limit behavior and the trend (monotonicity) when analyzing a sequence’s convergence and behavior over time.

Given the sequences an = (2n-1)/(4n) + (n-1)/(4n) and bn = 2n * an, evaluate whether the following statements are true or false.

Analyze the convergence, boundedness, and arithmetic progression of sequences an and bn. This detailed breakdown evaluates whether statements on monotonicity, limits, and boundedness are true, including core formulas and theorems related to sequences.

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