The question is asking to identify the sequence that has a common ratio of $-3$.

Let's analyze each option to determine the correct geometric sequence:

1. $1, 3, 9, 27, 81$

• The ratio between each term is $\frac{3}{1} = 3$, $\frac{9}{3} = 3$, etc.
• The common ratio is $3$, not $-3$.

2. $1, -3, 9, -27, 81$

• The ratio between each term is $\frac{-3}{1} = -3$, $\frac{9}{-3} = -3$, $\frac{-27}{9} = -3$.
• This has a common ratio of $-3$, so this is the correct sequence.

3. $9, 6, 3, 0, -3$

• The ratio between the terms is not consistent. The difference between terms is $-3$ in some steps, but there is no constant ratio.
• This is an arithmetic sequence, not geometric.

4. $9, 12, 15, 18, 21$

• This is an arithmetic sequence with a common difference of $3$, not a geometric sequence.

Conclusion:

The correct sequence with a common ratio of $-3$ is option 2: $1, -3, 9, -27, 81$.

Would you like further details or clarifications?

Here are 5 related questions for further practice:

1. What is the common ratio of the sequence $2, -6, 18, -54$?
2. How do you identify a geometric sequence from a list of numbers?
3. What is the sum of the first 5 terms of the sequence $1, -3, 9, -27, 81$?
4. How do geometric sequences differ from arithmetic sequences?
5. Can a geometric sequence have a ratio of $0$?

Tip: In a geometric sequence, the common ratio is found by dividing any term by the previous term.