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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Would you like further details on geometric sequences or any other clarification?

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!