What is the value of `1-2+3-4+5`- . . . .+101 ?

Options

A. 51
B. 55
C. 110
D. 111

Answer

To find the value of the series \(1 - 2 + 3 - 4 + 5 - 6 + \ldots + 101\), we can break it down into manageable parts. Here’s a step-by-step solution:

### Step 1: Group the terms
We can group the terms in pairs:
\[ (1 - 2) + (3 - 4) + (5 - 6) + \ldots + (99 - 100) + 101 \]

### Step 2: Calculate each pair
Each pair \( (n - (n+1)) \) results in \(-1\):
\[ 1 - 2 = -1, \quad 3 - 4 = -1, \quad 5 - 6 = -1, \ldots, \quad 99 - 100 = -1 \]

### Step 3: Count the pairs
From 1 to 100, there are 100 terms. Since we are pairing them, we have:
\[ \text{Number of pairs} = \frac{100}{2} = 50 \]
Thus, the contribution from these pairs is:
\[ 50 \times (-1) = -50 \]

### Step 4: Add the last term
Now, we need to add the last term, which is \(101\):
\[ -50 + 101 = 51 \]

### Final Answer
The value of the series \(1 - 2 + 3 - 4 + 5 - 6 + \ldots + 101\) is:
\[ \boxed{51} \]