i=1
∑
n
​

j=1
∑
i
​

k=j
∑
i+j
​
1

The expression you've written seems to involve a nested summation with three indices: \( i \), \( j \), and \( k \). To interpret this properly, let's break it down:

\[
\sum_{i=1}^{n} \sum_{j=1}^{i} \sum_{k=j}^{i+j} 1
\]

### Explanation:

1. **Outer Sum (\(i\))**: The outermost sum runs from \( i = 1 \) to \( i = n \).
2. **Middle Sum (\(j\))**: For each value of \( i \), the middle sum runs from \( j = 1 \) to \( j = i \).
3. **Inner Sum (\(k\))**: For each pair \( (i, j) \), the innermost sum runs from \( k = j \) to \( k = i+j \).

The expression inside the sums is simply 1, which means that for each valid combination of \( i \), \( j \), and \( k \), we add 1.

### Calculation:
The number of times 1 is added can be calculated by counting the total number of valid triples \( (i, j, k) \).

1. **For a fixed \( i \):**  
   - \( j \) runs from 1 to \( i \), giving \( i \) possible values for \( j \).
   - For each \( j \), \( k \) runs from \( j \) to \( i+j \), giving \( (i+j) - j + 1 = i+1 \) possible values for \( k \).

2. **Total Summation:**
   - The total number of 1's added for a fixed \( i \) is:
   \[
   \sum_{j=1}^{i} (i+1) = (i+1) \sum_{j=1}^{i} 1 = (i+1) \times i
   \]

3. **Outer Sum:**
   - The final sum over \( i \) from 1 to \( n \) is:
   \[
   \sum_{i=1}^{n} i(i+1) = \sum_{i=1}^{n} (i^2 + i)
   \]

   This can be split into:
   \[
   \sum_{i=1}^{n} i^2 + \sum_{i=1}^{n} i
   \]
   where:
   - \( \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \)
   - \( \sum_{i=1}^{n} i = \frac{n(n+1)}{2} \)

4. **Final Result:**
   \[
   \sum_{i=1}^{n} i(i+1) = \frac{n(n+1)(2n+1)}{6} + \frac{n(n+1)}{2}
   \]
   Simplifying this gives the total sum.

Would you like a detailed breakdown of each step, or do you have any specific questions about the process?

### Related Questions:
1. What is the value of the sum for specific small values of \( n \)?
2. How can the sum formula \( \sum_{i=1}^{n} i^2 \) be derived?
3. Can the sum be simplified further for large values of \( n \)?
4. How does the sum change if the limits of \( j \) and \( k \) are altered?
5. What is the relationship between the sum and triangular numbers?
6. How would the sum change if the innermost sum was a function of \( k \) instead of just 1?
7. What is the geometric interpretation of this summation process?
8. How do the sums relate to combinatorial problems?

**Tip:** Nested summations can often be simplified by carefully analyzing the limits of each sum and looking for patterns or known formulas to apply.

Explore the concept of nested summation in mathematics with detailed explanations and examples. This problem involves interpreting and calculating nested sums with multiple indices, providing insights into combinatorial sums and their applications. Suitable for advanced high school students and anyone interested in mathematical series and sums.

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