Exploring the Sum of a Specific Alternating Series: 1 - 2 3 - 4 5 - 6 7 - 8 9

Have you ever come across a math problem that seemed complicated at first but surprisingly had a simple and elegant solution? We'll delve into one such problem: the sum of the series 1 - 2 3 - 4 5 - 6 7 - 8 9. We will explore different methods to solve this series and the underlying principles that make this approach possible.

Method 1: Pairing Terms

The most straightforward approach to solving this series is by pairing the terms. Let's consider the series: 1 - 2 3 - 4 5 - 6 7 - 8 9.

We can group the terms in pairs as follows:

1 - 2 3 - 4 5 - 6 7 - 8 9

Calculating each pair:

1 - 2 -1 3 - 4 -1 5 - 6 -1 7 - 8 -1 9

Substituting back into the expression:

-1 - 1 - 1 - 1 9

Adding these together:

-1 - 1 - 1 - 1 9 -4 9 5

Conclusion

The sum of the series 1 - 2 3 - 4 5 - 6 7 - 8 9 is 5.

Method 2: Using Sequences and Formulas

Another way to solve this problem is by using the properties of arithmetic sequences and formulas. Let's break down the series step by step.

First, let's rewrite the series in a more formal manner:

1 - 2 3 - 4 5 - 6 7 - 8 9

This can be grouped as:

(1 - 2) (3 - 4) (5 - 6) (7 - 8) 9

Each pair of terms (1 - 2, 3 - 4, 5 - 6, 7 - 8) results in -1, and we have 4 such pairs. The last term, 9, remains unpaired.

Therefore, the sum can be expressed as:

-1 * 4 9 -4 9 5

General Case

Now, let's consider the general case for an alternating series of the form: 1 - 2 3 - 4 5 - 6 ... n.

We can divide the general case into two scenarios:

When n is an odd number: The sum would be -[n-1/2] * n. When n is an even number: The sum would be -n/2.

Conclusion

The sum of the series 1 - 2 3 - 4 5 - 6 7 - 8 9 is 5. Whether we use the pairing method or general formulas, the result remains consistent and elegant. This problem showcases the beauty and simplicity of mathematics in solving seemingly complex problems.