The sum of 100 terms of the series 1 - 3 + 5 - 7 + 9 - 11 + 13 - 15 + … is :

Options

A. 100
B. 50
C. 200
D. none of these

Answer

To find the sum of the first 100 terms of the series \(1 - 3 + 5 - 7 + 9 - 11 + 13 - 15 + \ldots\), we can break it down into two separate arithmetic progressions (APs).

### Step 1: Identify the two APs
The series can be split into two parts:
1. The positive terms: \(1, 5, 9, 13, \ldots\)
2. The negative terms: \(-3, -7, -11, -15, \ldots\)

### Step 2: Find the number of terms in each AP
Since there are 100 terms in total, there will be 50 positive terms and 50 negative terms.

### Step 3: Calculate the sum of the positive terms
The positive terms form an AP where:
- First term \(a = 1\)
- Common difference \(d = 4\)
- Number of terms \(n = 50\)

The sum \(S_n\) of the first \(n\) terms of an AP is given by the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Substituting the values:
\[ S_{positive} = \frac{50}{2} \times (2 \times 1 + (50 - 1) \times 4) \]
\[ = 25 \times (2 + 196) = 25 \times 198 = 4950 \]

### Step 4: Calculate the sum of the negative terms
The negative terms also form an AP where:
- First term \(a = -3\)
- Common difference \(d = -4\)
- Number of terms \(n = 50\)

Using the same formula:
\[ S_{negative} = \frac{50}{2} \times (2 \times (-3) + (50 - 1)(-4)) \]
\[ = 25 \times (-6 - 196) = 25 \times (-202) = -5050 \]

### Step 5: Calculate the total sum
Now, we combine the sums of the positive and negative terms:
\[ S_{total} = S_{positive} + S_{negative} = 4950 - 5050 = -100 \]

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

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