Let $n\ge 3$. Suppose $a_1, a_2, ... , a_n$ are $n$  distinct in pairs real numbers.
In terms of $n$, find the smallest possible number of different assumed values by the following $n$ numbers:
$$a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1$$
1. **Define the sequence and the sums:**
   Let \( a_1, a_2, \ldots, a_n \) be \( n \) distinct real numbers. We need to consider the sums \( b_i = a_i + a_{i+1} \) for \( i = 1, 2, \ldots, n \), where \( a_{n+1} = a_1 \). Thus, we have the sequence of sums:
   \[
   b_1 = a_1 + a_2, \quad b_2 = a_2 + a_3, \quad \ldots, \quad b_{n-1} = a_{n-1} + a_n, \quad b_n = a_n + a_1
   \]

2. **Establish the distinctness condition:**
   Since \( a_i \) are distinct, \( b_i \neq b_{i+1} \) for all \( i \). This is because if \( b_i = b_{i+1} \), then \( a_i + a_{i+1} = a_{i+1} + a_{i+2} \), which implies \( a_i = a_{i+2} \), contradicting the distinctness of \( a_i \).

3. **Consider the parity of \( n \):**
   - **For odd \( n \):** Since \( n \) is odd, the sequence \( b_1, b_2, \ldots, b_n \) must contain at least 3 distinct values. This is because if there were only 2 distinct values, say \( b_1 \) and \( b_2 \), they would alternate, leading to a contradiction when summing all \( a_i \).
   - **For even \( n \):** If \( n \) is even and we assume there are only 2 distinct values, say \( b_1 \) and \( b_2 \), then the sums would alternate as \( b_1, b_2, b_1, b_2, \ldots \). Summing all \( b_i \) would give:
     \[
     \sum_{i=1}^n b_i = \sum_{i=1}^n (a_i + a_{i+1}) = 2 \sum_{i=1}^n a_i
     \]
     If \( b_1 \) and \( b_2 \) are the only values, then:
     \[
     \frac{n}{2} b_1 + \frac{n}{2} b_2 = 2 \sum_{i=1}^n a_i
     \]
     This implies \( b_1 = b_2 \), which is a contradiction since \( b_i \neq b_{i+1} \).

4. **Construct an example with 3 distinct values:**
   Consider the sequence \( a_1 = 0, a_2 = 1, a_3 = -1, a_4 = 2, a_5 = -2, \ldots \). This sequence ensures that the sums \( b_i \) take on at least 3 distinct values.

5. **Conclusion:**
   Therefore, the smallest possible number of different assumed values by the sequence \( b_1, b_2, \ldots, b_n \) is 3.

The final answer is \(\boxed{3}\)