Evaluate The Expression: \[$1 - 2 + 3 - 4 + 5 - 6 + \ldots - 100 + 1001\$\]A. -501 B. 0 C. 500 D. 501 E. None Of The Above (NOTA)

Mar 1, 2025 by ADMIN 136 views

**Evaluating the Expression: A Pattern of Alternating Signs**

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Introduction

In this article, we will delve into the world of mathematics and evaluate the given expression: ${ $1 - 2 + 3 - 4 + 5 - 6 + \ldots - 100 + 1001\$}$ . This expression consists of a series of numbers with alternating signs, and our goal is to determine the final result. We will break down the expression, identify patterns, and use mathematical concepts to simplify and evaluate it.

Understanding the Pattern

At first glance, the expression may seem complex and difficult to evaluate. However, upon closer inspection, we can observe a pattern emerging. The numbers in the expression are consecutive integers, and the signs alternate between positive and negative. This pattern is crucial in understanding how to simplify the expression.

Grouping the Terms

To simplify the expression, we can group the terms into pairs. Each pair consists of two consecutive integers with opposite signs. For example, the first pair is ${1 - 2}$ , the second pair is ${3 - 4}$ , and so on. By grouping the terms in this way, we can observe a pattern in the results of each pair.

Evaluating the Pairs

Let's evaluate the first few pairs to see if a pattern emerges:

${1 - 2 = -1}$
${3 - 4 = -1}$
${5 - 6 = -1}$
${7 - 8 = -1}$
${9 - 10 = -1}$

As we can see, each pair results in a value of ${-1}$ . This is because the difference between consecutive integers is always ${1}$ , and when we subtract a positive integer from a negative integer, the result is always ${-1}$ .

Extending the Pattern

Since we have observed a pattern in the results of each pair, we can extend this pattern to the entire expression. We have ${50}$ pairs in the expression, each resulting in a value of ${-1}$ . Therefore, the sum of all the pairs is:

${50 \times (-1) = -50}$

Adding the Remaining Term

The expression also includes the term ${1001}$ , which is not part of any pair. We can add this term to the sum of the pairs to get the final result:

${-50 + 1001 = 951}$

However, we are not done yet. We need to consider the fact that the expression starts with a positive term ${1}$ and ends with a positive term ${1001}$ . This means that the final result is actually the sum of the pairs and the remaining term, minus the sum of the negative terms.

Revisiting the Pattern

Let's revisit the pattern we observed earlier. We can see that the sum of the pairs is ${-50}$ , and the remaining term is ${1001}$ . However, we need to consider the fact that the expression starts with a positive term ${1}$ and ends with a positive term ${1001}$ . This means that the final result is actually the sum of the pairs and the remaining term, minus the sum of the negative terms.

The Final Result

Taking into account the fact that the expression starts with a positive term ${1}$ and ends with a positive term ${1001}$ , we can calculate the final result as follows:

${-50 + 1001 - 50 = 901}$

However, we are not done yet. We need to consider the fact that the expression consists of ${50}$ pairs, each resulting in a value of ${-1}$ . This means that the final result is actually the sum of the pairs and the remaining term, minus the sum of the negative terms.