In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
1. **Define the problem and assumptions:**
   - Let \( a_1, a_2, \ldots, a_n \) be the sequence of real numbers.
   - The sum of any seven successive terms is negative:
     \[
     a_i + a_{i+1} + a_{i+2} + a_{i+3} + a_{i+4} + a_{i+5} + a_{i+6} < 0 \quad \text{for all } i
     \]
   - The sum of any eleven successive terms is positive:
     \[
     a_i + a_{i+1} + a_{i+2} + a_{i+3} + a_{i+4} + a_{i+5} + a_{i+6} + a_{i+7} + a_{i+8} + a_{i+9} + a_{i+10} > 0 \quad \text{for all } i
     \]

2. **Assume the sequence has 17 terms and derive a contradiction:**
   - Consider the sum of any 4 consecutive terms. We can express this as:
     \[
     (a_i + a_{i+1} + a_{i+2} + a_{i+3}) = (a_i + a_{i+1} + \cdots + a_{i+10}) - (a_{i+4} + a_{i+5} + a_{i+6} + a_{i+7} + a_{i+8} + a_{i+9} + a_{i+10})
     \]
     This is an 11-sum minus a 7-sum. Since the 11-sum is positive and the 7-sum is negative, the 4-sum must be positive:
     \[
     (a_i + a_{i+1} + a_{i+2} + a_{i+3}) > 0
     \]

3. **Consider the sum of any 3 consecutive terms:**
   - We can express this as:
     \[
     (a_i + a_{i+1} + a_{i+2}) = (a_i + a_{i+1} + a_{i+2} + a_{i+3}) - a_{i+3}
     \]
     Since the 4-sum is positive, and we subtract a single term, the 3-sum must be negative:
     \[
     (a_i + a_{i+1} + a_{i+2}) < 0
     \]

4. **Consider the value of a single term:**
   - We can express a single term as:
     \[
     a_i = (a_i + a_{i+1} + a_{i+2} + a_{i+3}) - (a_{i+1} + a_{i+2} + a_{i+3})
     \]
     Since the 4-sum is positive and the 3-sum is negative, the single term must be positive:
     \[
     a_i > 0
     \]

5. **Reach a contradiction:**
   - If every term \( a_i \) is positive, then the sum of any 7 consecutive terms cannot be negative, which contradicts the given condition.

6. **Conclusion:**
   - Therefore, the sequence cannot have 17 terms. The maximum number of terms in the sequence must be 16.

The final answer is \( \boxed{16} \)