3) Determine whether each sequence is arithmetic or geometricor otherwise. If it is a geometric sequence, find the common ratio.

c) -1, 6, -36, 216, ...

Let's analyze the given sequence: \(-1, 6, -36, 216, \ldots\).

### Step 1: Check if the sequence is geometric
To determine if the sequence is geometric, we need to check if there is a common ratio between consecutive terms. A sequence is geometric if the ratio between consecutive terms remains constant.

We can find the ratio \( r \) by dividing any term by its previous term:

1. \( r_1 = \frac{6}{-1} = -6 \)
2. \( r_2 = \frac{-36}{6} = -6 \)
3. \( r_3 = \frac{216}{-36} = -6 \)

Since the ratio between consecutive terms is consistently \(-6\), the sequence is indeed **geometric**.

### Step 2: Common ratio
The common ratio \( r \) is \(-6\).

### Conclusion
The sequence \(-1, 6, -36, 216, \ldots\) is a **geometric sequence** with a common ratio of \(-6\).

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Would you like more details or have any questions?

Here are 5 related questions:
1. How do you identify an arithmetic sequence?
2. What happens if the common ratio in a geometric sequence is a fraction?
3. How can you derive a general formula for the \(n\)-th term of a geometric sequence?
4. Can a sequence be neither arithmetic nor geometric? How do you classify such sequences?
5. How does changing the first term affect the overall behavior of a geometric sequence?

**Tip:** When working with geometric sequences, always check the ratio between multiple consecutive terms to confirm it's consistent throughout the sequence.

Determine whether each sequence is arithmetic or geometric, or otherwise. If it is a geometric sequence, find the common ratio. c) -1, 6, -36, 216, ...

Learn how to determine if the sequence -1, 6, -36, 216 is arithmetic, geometric, or neither. This problem involves checking for a common ratio and identifying the geometric sequence with a common ratio of -6. Suitable for students in Grades 9-12.

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