To determine the next three terms of the geometric sequence with the given first term and common ratio, we will follow these steps:
1. Identify the first term and the common ratio:
- First term ([tex]\(a_1\)[/tex]) = -2
- Common ratio ([tex]\(r\)[/tex]) = -[tex]\(\frac{1}{4}\)[/tex]
2. Calculate the second term ([tex]\(a_2\)[/tex]):
- [tex]\(a_2 = a_1 \times r = -2 \times -\frac{1}{4}\)[/tex]
- [tex]\(a_2 = \frac{1}{2}\)[/tex]
3. Calculate the third term ([tex]\(a_3\)[/tex]):
- [tex]\(a_3 = a_2 \times r = \frac{1}{2} \times -\frac{1}{4}\)[/tex]
- [tex]\(a_3 = -\frac{1.}{8}\)[/tex]
4. Calculate the fourth term ([tex]\(a_4\)[/tex]):
- [tex]\(a_4 = a_3 \times r = -\frac{1}{8} \times -\frac{1}{4}\)[/tex]
- [tex]\(a_4 = \frac{1}{32}\)[/tex]
So, the next three terms of the sequence would be:
[tex]\[
\frac{1}{2}, -\frac{1}{8}, \text{ and } \frac{1}{32}
\][/tex]
Let's match these with the provided answer choices:
- [tex]\(-\frac{1}{2},-\frac{1}{8},-\frac{1}{32}\)[/tex]
- [tex]\(\frac{1}{2},-\frac{1}{8}, \cdot \frac{1}{32}\)[/tex]
- [tex]\(-\frac{1}{2}, \frac{1}{8},-\frac{1}{32}\)[/tex]
- [tex]\(\frac{1}{2}, \frac{1}{8}, \frac{1}{32}\)[/tex]
The correct set of next three terms is:
[tex]\[
\frac{1}{2}, -\frac{1}{8}, \frac{1}{32}
\][/tex]
