4. What is the biggest number of terms for which... \( \sum_{r=1}^{n}(r+1) \)

To find the biggest number of terms for the sum \( \sum_{r=1}^{n}(r+1) \), we can first simplify the expression. 1. **Rewrite the sum**: \[ \sum_{r=1}^{n}(r+1) = \sum_{r=1}^{n} r + \sum_{r=1}^{n} 1 \] 2. **Calculate each part**: - The first part, \( \sum_{r=1}^{n} r \), is the sum of the first \( n \) natural numbers, which can be calculated using the formula: \[ \sum_{r=1}^{n} r = \frac{n(n+1)}{2} \] - The second part, \( \sum_{r=1}^{n} 1 \), simply counts the number of terms, which is \( n \). 3. **Combine the results**: \[ \sum_{r=1}^{n}(r+1) = \frac{n(n+1)}{2} + n \] 4. **Simplify further**: \[ \sum_{r=1}^{n}(r+1) = \frac{n(n+1)}{2} + \frac{2n}{2} = \frac{n(n+1) + 2n}{2} = \frac{n^2 + 3n}{2} \] Now, we have the expression for the sum \( \sum_{r=1}^{n}(r+1) = \frac{n^2 + 3n}{2} \). To find the biggest number of terms, we need to determine the maximum value of \( n \) for which this sum is defined. Since \( n \) is a positive integer, the biggest number of terms is simply the largest integer \( n \) that we can consider. Thus, the answer is that the biggest number of terms \( n \) can be any positive integer, and there is no upper limit defined in the problem. Therefore, \( n \) can be as large as desired, theoretically approaching infinity.