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.