https://r2.erweima.ai/mathbot/image/1726477599985yj2iq2yt.jpg

The question in the image is asking to find the value of the series:

\[
S = -1 + 2 - 3 + 4 - 5 + 6 - \cdots + 99 + 100
\]

### Solution:
This is an alternating series, where odd numbers are negative and even numbers are positive. To simplify the calculation, group the terms two by two:

\[
(-1 + 2) + (-3 + 4) + (-5 + 6) + \cdots + (-99 + 100)
\]

Each pair of terms simplifies as follows:

\[
(-1 + 2) = 1,\ (-3 + 4) = 1,\ (-5 + 6) = 1,\ \cdots,\ (-99 + 100) = 1
\]

There are 50 such pairs (since the numbers run from 1 to 100), so the total sum is:

\[
S = 50 \times 1 = 50
\]

Thus, the value of the series is **50**.

### The correct answer is option **D) 50**.

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Do you need more details or have any questions?

Here are 5 related questions you might want to explore:
1. How do you sum an arithmetic series with alternating signs?
2. How can you quickly identify patterns in number series?
3. What are some common techniques for summing series in calculus?
4. How can grouping terms in a series simplify calculations?
5. What is the general formula for summing an arithmetic progression?

**Tip:** For alternating series, pairing terms often helps reveal a simpler structure, making the summation more manageable.

求 -1 + 2 - 3 + 4 - 5 + 6 ... - 99 + 100 之值為何?

Find the value of the arithmetic series with alternating signs: -1 + 2 - 3 + 4 ... - 99 + 100. This solution demonstrates how to group terms in pairs to simplify the summation process, resulting in the total sum of 50. Ideal for students in Grades 9-10.

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