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Let's analyze and solve the given problem step by step.

**Problem Statement:**

Given a sequence \( \{a_n\} \), where \( a_1 = 1 \), and the points \( P(a_n, 2a_{n+1}) \) lie on the line \( x - \frac{1}{2}y + 1 = 0 \).

1. Find the general term formula for the sequence \( \{a_n\} \).
2. Find the minimum value of the function 
\[ f(n) = \frac{1}{n + a_1} + \frac{1}{n + a_2} + \frac{1}{n + a_3} + \cdots + \frac{1}{n + a_n} \]
   for \( n \geq 2 \).

### Step-by-Step Solution:

#### Part (1): General Term Formula for \( \{a_n\} \)

The given condition is that the points \( P(a_n, 2a_{n+1}) \) lie on the line \( x - \frac{1}{2}y + 1 = 0 \).

Substituting \( P(a_n, 2a_{n+1}) \) into the line equation:
\[ a_n - \frac{1}{2}(2a_{n+1}) + 1 = 0 \]
\[ a_n - a_{n+1} + 1 = 0 \]
\[ a_{n+1} = a_n + 1 \]

Given \( a_1 = 1 \), we can see that \( a_n \) is an arithmetic sequence where:
\[ a_{n+1} = a_n + 1 \]

The general term \( a_n \) for an arithmetic sequence with the first term \( a_1 = 1 \) and a common difference of 1 is:
\[ a_n = 1 + (n-1) = n \]

So, the general term is:
\[ a_n = n \]

#### Part (2): Minimum Value of \( f(n) \)

The function \( f(n) \) is given by:
\[ f(n) = \frac{1}{n + a_1} + \frac{1}{n + a_2} + \cdots + \frac{1}{n + a_n} \]

Substituting \( a_i = i \), we get:
\[ f(n) = \frac{1}{n + 1} + \frac{1}{n + 2} + \cdots + \frac{1}{n + n} \]
\[ f(n) = \sum_{k=1}^n \frac{1}{n+k} \]

This sum is decreasing as \( n \) increases because each term \( \frac{1}{n+k} \) gets smaller. Hence, to find the minimum value of \( f(n) \), we analyze it for the smallest \( n \geq 2 \):

For \( n = 2 \):
\[ f(2) = \frac{1}{2 + 1} + \frac{1}{2 + 2} = \frac{1}{3} + \frac{1}{4} = \frac{4 + 3}{12} = \frac{7}{12} \]

As \( n \) increases, \( f(n) \) will decrease because the terms \( \frac{1}{n+k} \) will become smaller. Hence, the minimum value of \( f(n) \) when \( n \geq 2 \) occurs at \( n = 2 \) and is:
\[ \boxed{\frac{7}{12}} \]

### Conclusion:

1. The general term of the sequence \( \{a_n\} \) is \( a_n = n \).
2. The minimum value of the function \( f(n) \) for \( n \geq 2 \) is \( \frac{7}{12} \).

Would you like any additional details or have any questions about this solution?

Here are five questions you might find interesting:
1. How do you determine the common difference in an arithmetic sequence?
2. What is the general formula for the sum of an arithmetic series?
3. Can you explain how to derive the equation of a line given two points?
4. How does the harmonic series relate to the sum we calculated in \( f(n) \)?
5. What are some applications of finding the minimum value of a function in real-world problems?

**Tip:** Understanding the behavior of sequences and series is crucial in calculus, especially for solving problems involving limits and summations.

Explore the analysis and solution steps for determining the general term formula of an arithmetic sequence and finding the minimum value of a specific function. This problem involves arithmetic sequences, series summation, and line equation applications. Suitable for high school students.

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