Write the next three terms of the following sequence.

512, -256, 128, -64, . . .

Group of answer choices

-32, 16, and -8

128, -256, and 512

-128, 256, and -512

32, -16, and 8

Cannot be determined.

This sequence alternates between positive and negative terms, and it appears to follow a geometric progression where each term is multiplied by \(-\frac{1}{2}\) to get the next term.

Let's verify by dividing each consecutive term by the previous one:

\[
\frac{-256}{512} = -\frac{1}{2}
\]
\[
\frac{128}{-256} = -\frac{1}{2}
\]
\[
\frac{-64}{128} = -\frac{1}{2}
\]

Since the common ratio is \(-\frac{1}{2}\), we can find the next three terms by multiplying the last term, \(-64\), by \(-\frac{1}{2}\) repeatedly:

1. \(-64 \times -\frac{1}{2} = 32\)
2. \(32 \times -\frac{1}{2} = -16\)
3. \(-16 \times -\frac{1}{2} = 8\)

Thus, the next three terms are: **32, -16, and 8**.

The correct answer is:
**32, -16, and 8**.

Would you like further clarification or details on this sequence? Here are 5 related questions to expand your understanding:

1. What is the general formula for the nth term of a geometric sequence?
2. How do you determine the common ratio of a geometric sequence?
3. Can a geometric sequence have a positive common ratio?
4. How would the sequence change if the common ratio were positive?
5. How do you determine the sum of an infinite geometric series?

**Tip:** In a geometric sequence, the ratio between consecutive terms remains constant.

Write the next three terms of the following sequence: 512, -256, 128, -64, . . . Group of answer choices: -32, 16, and -8; 128, -256, and 512; -128, 256, and -512; 32, -16, and 8; Cannot be determined.

Discover how to find the next three terms in the geometric sequence 512, -256, 128, -64, using the concept of geometric progression with a negative common ratio. This problem is ideal for students in Grades 8-10, focusing on the geometric progression theorem and negative common ratios.

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