The additive inverse of S, where `S = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + …. + 49 - 50` is

Options

A. `-25`
B. 1
C. 0
D. 25

Answer

To find the additive inverse of \( S \), where \( S = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + \ldots + 49 - 50 \), we will first calculate the value of \( S \).

### Step 1: Identify the pattern in \( S \)
The expression alternates between adding and subtracting consecutive integers:
- The first term is positive (1), the second term is negative (2), the third term is positive (3), and so on, up to 50.

### Step 2: Group the terms
We can group the terms into pairs:
- \( (1 - 2), (3 - 4), (5 - 6), \ldots, (49 - 50) \)

### Step 3: Calculate the value of each pair
Each pair can be calculated as follows:
- \( 1 - 2 = -1 \)
- \( 3 - 4 = -1 \)
- \( 5 - 6 = -1 \)
- ...
- \( 49 - 50 = -1 \)

### Step 4: Count the number of pairs
From 1 to 50, there are 50 numbers. Since we are pairing them, we have:
- Total pairs = \( \frac{50}{2} = 25 \)

### Step 5: Calculate the total value of \( S \)
Since each of the 25 pairs sums to -1:
- Total value of \( S = 25 \times (-1) = -25 \)

### Step 6: Find the additive inverse of \( S \)
The additive inverse of a number \( x \) is the number that, when added to \( x \), results in zero. Therefore, the additive inverse of \( S \) is:
- If \( S = -25 \), then the additive inverse is \( 25 \) (since \( -25 + 25 = 0 \)).

### Conclusion
Thus, the additive inverse of \( S \) is \( 25 \).