Which sequence is geometric? -2,4,-6,-10           1,1,2,3,5,8.       12,7,2,-3,-8.      -27,-9,-3,-1,-1/3

A **geometric sequence** is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the **common ratio**. Let's check each sequence:

1. **Sequence**: -2, 4, -6, -10  
   To find the common ratio, divide each term by the previous term:
   - \( \frac{4}{-2} = -2 \)
   - \( \frac{-6}{4} = -\frac{3}{2} \)
   - \( \frac{-10}{-6} = \frac{5}{3} \)
   
   The ratio between terms is not constant, so this is **not a geometric sequence**.

2. **Sequence**: 1, 1, 2, 3, 5, 8  
   - \( \frac{1}{1} = 1 \)
   - \( \frac{2}{1} = 2 \)
   - \( \frac{3}{2} \neq 2 \)
   
   Since the ratio is not constant, this is **not a geometric sequence**.

3. **Sequence**: 12, 7, 2, -3, -8  
   - \( \frac{7}{12} \neq \frac{2}{7} \)
   
   This sequence also does not have a constant ratio, so it is **not geometric**.

4. **Sequence**: -27, -9, -3, -1, -1/3  
   - \( \frac{-9}{-27} = \frac{1}{3} \)
   - \( \frac{-3}{-9} = \frac{1}{3} \)
   - \( \frac{-1}{-3} = \frac{1}{3} \)
   - \( \frac{-\frac{1}{3}}{-1} = \frac{1}{3} \)
   
   The ratio is constant (\( \frac{1}{3} \)) throughout, so this is a **geometric sequence**.

### Conclusion:
The geometric sequence is **-27, -9, -3, -1, -1/3**.

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### Related questions:
1. What is the formula for the nth term of a geometric sequence?
2. How do you find the sum of a finite geometric series?
3. What is the difference between arithmetic and geometric sequences?
4. Can a geometric sequence have a negative common ratio?
5. How do you calculate the sum of an infinite geometric series?

**Tip**: In a geometric sequence, dividing any term by the previous term always gives the same result—the common ratio!

Learn how to identify geometric sequences from a list of sequences. This problem involves recognizing sequences with a constant common ratio and covers essential concepts in algebra for students in Grades 8-10.

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