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.