Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.
To solve the problem, we need to find all possible values of \( n \) such that \( \frac{1}{n} = 0.a_1a_2a_3\ldots \) and \( n = a_1 + a_2 + a_3 + \cdots \).

1. **Express \( n \) in terms of its prime factors:**
   Since \( \frac{1}{n} \) is a terminating decimal, \( n \) must be of the form \( n = 2^a \cdot 5^b \) for some non-negative integers \( a \) and \( b \).

2. **Consider the sum of the digits of \( \frac{1}{n} \):**
   Let \( \frac{1}{n} = 0.a_1a_2a_3\ldots \) where \( a_1, a_2, a_3, \ldots \) are the digits of the decimal expansion. We are given that \( n = a_1 + a_2 + a_3 + \cdots \).

3. **Analyze the possible values of \( n \):**
   We need to check the possible values of \( n \) by considering the constraints on \( a \) and \( b \).

   - **Case 1: \( k = a = b \):**
     If \( k = a = b \), then \( n = 2^k \cdot 5^k = 10^k \). For \( k \ge 1 \), \( 10^k \) is always greater than the sum of its digits. For example, if \( k = 1 \), \( n = 10 \) and the sum of the digits of \( \frac{1}{10} = 0.1 \) is 1, which is not equal to 10. Hence, this case leads to a contradiction.

   - **Case 2: \( k = a > b \):**
     If \( k = a > b \), then \( n = 2^a \cdot 5^b \) where \( a > b \). We need to check if there are any values of \( a \) and \( b \) such that the sum of the digits of \( \frac{1}{n} \) equals \( n \).

     For example, if \( a = 3 \) and \( b = 0 \), then \( n = 2^3 = 8 \). The decimal expansion of \( \frac{1}{8} = 0.125 \) and the sum of the digits is \( 1 + 2 + 5 = 8 \), which matches \( n \). Therefore, \( n = 8 \) is a solution.

   - **Case 3: \( k = b > a \):**
     If \( k = b > a \), then \( n = 2^a \cdot 5^b \) where \( b > a \). We need to check if there are any values of \( a \) and \( b \) such that the sum of the digits of \( \frac{1}{n} \) equals \( n \).

     For example, if \( a = 0 \) and \( b = 1 \), then \( n = 5 \). The decimal expansion of \( \frac{1}{5} = 0.2 \) and the sum of the digits is \( 2 \), which does not match \( n \). Hence, there are no solutions in this case.

4. **Conclusion:**
   The only value of \( n \) that satisfies the given conditions is \( n = 8 \).

The final answer is \( \boxed{8} \).