To find the next term in the sequence $1, 3, -9, -27, 81$, we need to identify the pattern or rule governing the sequence.

Step 1: Identify the pattern

Let's look at how each term is related to the previous one.

1. From 1 to 3: $3 = 1 \times 3$
2. From 3 to -9: $-9 = 3 \times -3$
3. From -9 to -27: $-27 = -9 \times 3$
4. From -27 to 81: $81 = -27 \times -3$

The pattern alternates between multiplying by 3 and by -3:

• $1 \times 3 = 3$
• $3 \times -3 = -9$
• $-9 \times 3 = -27$
• $-27 \times -3 = 81$

Step 2: Apply the pattern to find the next term

Following the pattern, the next operation would involve multiplying by 3 again (since the pattern alternates): $81 \times 3 = 243$

Conclusion:

The next term in the sequence is 243.

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Do you want more details, or have any questions about the solution? Here are 8 related questions you might consider:

1. What is a geometric sequence, and how does it differ from this sequence?
2. How can you identify the general term of a sequence?
3. What is the formula for the nth term of an alternating sequence?
4. How do you determine if a sequence is arithmetic, geometric, or neither?
5. Can you derive a formula for the nth term of this specific sequence?
6. How does changing the sign in the pattern affect the sequence?
7. What other types of sequences follow an alternating sign pattern?
8. How can you use sequences to model real-world problems?

Tip: When solving sequences, always check if there's a common ratio or difference, as this can help quickly identify the type of sequence you're dealing with.