Find `sum_(i=1)^n sum_(i=1)^n sum_(k=1)^n (ijk)`

Answer

To find the value of the summation \( \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^n (ijk) \), we can break it down step by step.

### Step 1: Understand the Summation
We need to evaluate the triple summation where \( i, j, k \) each run from 1 to \( n \). This means we will be summing the product \( ijk \) for all combinations of \( i, j, k \).

### Step 2: Separate the Summation
We can separate the summation into individual parts. The expression can be rewritten as:
\[ \sum_{i=1}^n i \sum_{j=1}^n j \sum_{k=1}^n k \]

### Step 3: Calculate Each Summation
Now we need to calculate each of the summations \( \sum_{i=1}^n i \), \( \sum_{j=1}^n j \), and \( \sum_{k=1}^n k \). The formula for the sum of the first \( n \) natural numbers is:
\[ \sum_{m=1}^n m = \frac{n(n+1)}{2} \]
Thus, we have:
\[ \sum_{i=1}^n i = \frac{n(n+1)}{2} \]
\[ \sum_{j=1}^n j = \frac{n(n+1)}{2} \]
\[ \sum_{k=1}^n k = \frac{n(n+1)}{2} \]

### Step 4: Combine the Results
Now we can substitute these results back into our separated summation:
\[ \sum_{i=1}^n i \sum_{j=1}^n j \sum_{k=1}^n k = \left( \frac{n(n+1)}{2} \right) \left( \frac{n(n+1)}{2} \right) \left( \frac{n(n+1)}{2} \right) \]

### Step 5: Simplify the Expression
This simplifies to:
\[ = \left( \frac{n(n+1)}{2} \right)^3 = \frac{n^3 (n+1)^3}{8} \]

### Final Result
Thus, the final result for the summation \( \sum_{i=1}^n \sum_{j=1}^n \sum_{k=1}^n (ijk) \) is:
\[ \frac{n^3 (n+1)^3}{8} \]