Math Problem Statement

i=1 ∑ n ​

j=1 ∑ i ​

k=j ∑ i+j ​ 1

Solution

The expression you've written seems to involve a nested summation with three indices: ii, jj, and kk. To interpret this properly, let's break it down:

i=1nj=1ik=ji+j1\sum_{i=1}^{n} \sum_{j=1}^{i} \sum_{k=j}^{i+j} 1

Explanation:

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

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

  1. For a fixed ii:

    • jj runs from 1 to ii, giving ii possible values for jj.
    • For each jj, kk runs from jj to i+ji+j, giving (i+j)j+1=i+1(i+j) - j + 1 = i+1 possible values for kk.
  2. Total Summation:

    • The total number of 1's added for a fixed ii is: j=1i(i+1)=(i+1)j=1i1=(i+1)×i\sum_{j=1}^{i} (i+1) = (i+1) \sum_{j=1}^{i} 1 = (i+1) \times i
  3. Outer Sum:

    • The final sum over ii from 1 to nn is: i=1ni(i+1)=i=1n(i2+i)\sum_{i=1}^{n} i(i+1) = \sum_{i=1}^{n} (i^2 + i)

    This can be split into: i=1ni2+i=1ni\sum_{i=1}^{n} i^2 + \sum_{i=1}^{n} i where:

    • i=1ni2=n(n+1)(2n+1)6\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}
    • i=1ni=n(n+1)2\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
  4. Final Result: i=1ni(i+1)=n(n+1)(2n+1)6+n(n+1)2\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 nn?
  2. How can the sum formula i=1ni2\sum_{i=1}^{n} i^2 be derived?
  3. Can the sum be simplified further for large values of nn?
  4. How does the sum change if the limits of jj and kk 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 kk 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.

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Math Problem Analysis

Mathematical Concepts

Nested summation
Combinatorial sums

Formulas

Summation formula for nested indices

Theorems

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Suitable Grade Level

Advanced High School