https://r2.erweima.ai/mathbot/image/172229179993225fh25a3.png

Let's analyze each sequence to determine if it is geometric, and if it is, find the common ratio.

### (a) Sequence: 4, 1, -1, -4, ...

To determine if the sequence is geometric, we need to check if there is a common ratio \( r \) between consecutive terms.

\[
r = \frac{\text{second term}}{\text{first term}} = \frac{1}{4}
\]

\[
r = \frac{\text{third term}}{\text{second term}} = \frac{-1}{1} = -1
\]

Since \( \frac{1}{4} \neq -1 \), the sequence is not geometric.

### (b) Sequence: 4, 2, 1, \(\frac{1}{2}\), ...

Check the common ratio \( r \):

\[
r = \frac{\text{second term}}{\text{first term}} = \frac{2}{4} = \frac{1}{2}
\]

\[
r = \frac{\text{third term}}{\text{second term}} = \frac{1}{2} = \frac{1}{2}
\]

\[
r = \frac{\text{fourth term}}{\text{third term}} = \frac{\frac{1}{2}}{1} = \frac{1}{2}
\]

Since the ratio is consistent, the sequence is geometric with a common ratio \( r = \frac{1}{2} \).

### (c) Sequence: 25, 20, 15, 10, ...

Check the common ratio \( r \):

\[
r = \frac{\text{second term}}{\text{first term}} = \frac{20}{25} = \frac{4}{5}
\]

\[
r = \frac{\text{third term}}{\text{second term}} = \frac{15}{20} = \frac{3}{4}
\]

\[
r = \frac{\text{fourth term}}{\text{third term}} = \frac{10}{15} = \frac{2}{3}
\]

Since the ratios are not consistent, the sequence is not geometric.

### Summary:

- (a) Not geometric
- (b) Geometric, common ratio \( r = \frac{1}{2} \)
- (c) Not geometric

Would you like more details or have any questions?

Here are 8 related questions:

1. How do you determine if a sequence is geometric?
2. What is a common ratio in a geometric sequence?
3. Can a geometric sequence have negative terms?
4. How can you find the nth term of a geometric sequence?
5. What is the sum of the first n terms of a geometric sequence?
6. Can a geometric sequence have a common ratio of 1?
7. What is an example of a real-life application of geometric sequences?
8. How do geometric sequences differ from arithmetic sequences?

**Tip:** Always check the ratio between consecutive terms when identifying geometric sequences. Consistency is key!

Explore geometric sequences and learn how to identify common ratios between terms. This analysis covers examples and detailed explanations to determine if a sequence is geometric. Suitable for high school students and anyone interested in advanced mathematical patterns.

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