Problem 116 - Elementary and Intermediate Algebra

Question

Find: $$1-2+3-4+5-6+\ldots+99-100$$

Step-by-Step Solution

Answer

The sum of the series is -50.

Step 1: Understanding the Series

The series is an alternation of positive and negative integers from 1 to 100. Specifically, odd numbers appear with a positive sign and even numbers with a negative sign. We need to determine the sum of this entire series.

Step 2: Pairing the Terms

Notice that the series can be paired as follows: \((1 - 2), (3 - 4), (5 - 6), \ldots, (99 - 100)\). This creates 50 pairs in total, each contributing to the final sum.

Step 3: Calculating Each Pair

Each pair consists of two consecutive numbers with the form \((n - (n+1))\), which simplifies to \(-1\). So, each pair sums to \(-1\).

Step 4: Summing Up All Pairs

Since there are 50 pairs and each pair sums to \(-1\), the total sum of the series is calculated as \(50 \times (-1) = -50\).

Key Concepts

Alternating Series

An alternating series is a series where the sign of each term alternates between positive and negative as we progress through the sequence. In this particular exercise, the series starts with a positive term and alternates as

• positive (+1),
• negative (-2),
• positive (+3),
• negative (-4),

and so forth. Alternating series are interesting because they can converge (approach a finite limit) even when their individual terms do not tend toward zero. However, in this example, since the numbers are being added and subtracted in pairs and do not reach zero individually, the series converges to a finite number based on the calculation of paired terms. By understanding the structure as alternating, we can intelligently reorder to simplify the sum, like pairing consecutive positive and negative terms.

Series Summation

Summation of a series involves finding the total of all its terms. For this arithmetic series, each pair of terms should be evaluated and then summed up. Each of these pairs, like \(1 - 2\), naturally simplifies to -1. The process makes it simple to find the sum because instead of summing 100 numbers individually, we simplify to 50 calculations of identical pairs.

Finding a pattern in series summation can help in achieving a solution quickly. For the series \(1-2+3-4+\ldots+99-100\), recognizing every successive pair equating to -1 allows swift calculation by multiplying the number of pairs by -1. This efficiency not only speeds up the calculation but also helps in avoiding errors that can arise from long manual additions or subtractions.

Consecutive Numbers

Consecutive numbers are numbers that follow each other in order, with no gaps. In the series from 1 to 100, each number is followed directly by its successor. These numbers maintain a predictable pattern, allowing them to be paired easily: \(1, 2\), \(3, 4\), and so on.

When handling arithmetic problems with consecutive numbers, pairing them strategically is a helpful technique. In this exercise, consecutive numbers, with different signs add up to -1 within each pair, simplifying the overall arithmetic process. Understanding this characteristic of consecutive numbers not only clarifies the approach to solving such series but also streamlines the calculation process, ensuring accurate results with minimal effort. Identifying patterns in the series helps in crafting solutions methodically and logically.