What is the sum of 100 terms of the series
`1-2+3-4+5-6+…. `?

Options

A. 100
B. 50
C. 550
D. none of these

Answer

To find the sum of the first 100 terms of the series \(1 - 2 + 3 - 4 + 5 - 6 + \ldots\), we can follow these steps:

### Step 1: Understand the Pattern
The series alternates between positive and negative integers. We can group the terms in pairs:
\[ (1 - 2), (3 - 4), (5 - 6), \ldots \]

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

### Step 3: Determine the Number of Pairs
Since we are looking for the sum of the first 100 terms, we can see that we have 50 pairs (because \(100 \div 2 = 50\)).

### Step 4: Calculate the Total Sum
Now, we can calculate the total sum of these pairs:
\[ \text{Total Sum} = 50 \times (-1) = -50 \]

### Final Answer
Therefore, the sum of the first 100 terms of the series is \(-50\).

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