What is te value of -1 + 2 - 3 + 4 - 5 + 6 - 7 + … + 1000 ?

Options

A. 500
B. 2000
C. 0
D. 1

Answer

To solve the expression \(-1 + 2 - 3 + 4 - 5 + 6 - 7 + \ldots + 1000\), we can follow these steps:

### Step 1: Identify the Pattern
The series alternates between negative and positive numbers. The negative terms are odd numbers and the positive terms are even numbers.

### Step 2: Group the Terms
We can group the terms in pairs:
\[ (-1 + 2) + (-3 + 4) + (-5 + 6) + \ldots + (-999 + 1000) \]
Each pair consists of one negative and one positive number.

### Step 3: Calculate Each Pair
Each pair simplifies to:
\[ (-n + (n+1)) = 1 \]
For example:
\[ (-1 + 2) = 1, \quad (-3 + 4) = 1, \quad (-5 + 6) = 1, \quad \ldots \]

### Step 4: Count the Number of Pairs
Since we are going from \(-1\) to \(1000\), we have a total of \(1000\) terms. The pairs are formed by taking two terms at a time, so the number of pairs is:
\[ \frac{1000}{2} = 500 \]

### Step 5: Calculate the Total Sum
Since each pair sums to \(1\) and there are \(500\) pairs, the total sum is:
\[ 500 \times 1 = 500 \]

### Final Answer
Thus, the value of the series \(-1 + 2 - 3 + 4 - 5 + 6 - 7 + \ldots + 1000\) is:
\[ \boxed{500} \]

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