`sum_(i=1)^(n) sum_(i=1)^(n) i` is equal to

Options

A. `(n(n+1))/(2)`
B. `(n(n+1)^(2))/2`
C. `(n^(2)(n+1))/2`
D. none of these

Answer

To solve the problem \( \sum_{i=1}^{n} \sum_{j=1}^{n} j \), we can follow these steps:

### Step 1: Understand the Inner Summation
The inner summation \( \sum_{j=1}^{n} j \) represents the sum of the first \( n \) natural numbers. We can express this sum using the formula:
\[ \sum_{j=1}^{n} j = \frac{n(n + 1)}{2} \]

### Step 2: Substitute the Inner Summation
Now, we substitute the result of the inner summation back into the outer summation:
\[ \sum_{i=1}^{n} \sum_{j=1}^{n} j = \sum_{i=1}^{n} \left( \frac{n(n + 1)}{2} \right) \]

### Step 3: Simplify the Outer Summation
Since \( \frac{n(n + 1)}{2} \) is a constant with respect to \( i \), we can factor it out of the outer summation:
\[ \sum_{i=1}^{n} \left( \frac{n(n + 1)}{2} \right) = \frac{n(n + 1)}{2} \sum_{i=1}^{n} 1 \]

### Step 4: Evaluate the Remaining Summation
The summation \( \sum_{i=1}^{n} 1 \) simply counts the number of terms from 1 to \( n \), which is \( n \):
\[ \sum_{i=1}^{n} 1 = n \]

### Step 5: Combine the Results
Now, we can combine our results:
\[ \frac{n(n + 1)}{2} \cdot n = \frac{n^2(n + 1)}{2} \]

### Final Answer
Thus, the value of \( \sum_{i=1}^{n} \sum_{j=1}^{n} j \) is:
\[ \frac{n^2(n + 1)}{2} \]

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