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

Options

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

Answer

To find the additive inverse of \( S \), where \( S = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + \ldots + 49 - 50 \), we will follow these steps:

### Step 1: Identify the pattern in the series
The series alternates between adding and subtracting consecutive integers:
\[ S = (1 - 2) + (3 - 4) + (5 - 6) + (7 - 8) + \ldots + (49 - 50) \]

### Step 2: Calculate each pair
Each pair can be simplified:
\[ 1 - 2 = -1, \quad 3 - 4 = -1, \quad 5 - 6 = -1, \quad 7 - 8 = -1, \ldots \]
Thus, each pair contributes \(-1\) to the sum.

### Step 3: Count the number of pairs
The numbers from 1 to 50 consist of 50 integers. Since we are pairing them, we have:
\[ \text{Number of pairs} = \frac{50}{2} = 25 \]

### Step 4: Calculate the total sum \( S \)
Since each of the 25 pairs contributes \(-1\), we have:
\[ S = 25 \times (-1) = -25 \]

### Step 5: Find the additive inverse of \( S \)
The additive inverse of a number \( x \) is defined as \(-x\). Therefore, the additive inverse of \( S \) is:
\[ \text{Additive Inverse of } S = -(-25) = 25 \]

### Final Answer
The additive inverse of \( S \) is \( 25 \).

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