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The problem asks which of the given sequences is a geometric sequence and to find the common ratio for each geometric sequence.

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant value, known as the common ratio (\(r\)).

### Let's analyze each sequence to check if it's geometric and find the common ratio if it is:

1. **Option a: \( \frac{1}{2}, 1, 2, 4, \dots \)**
   - Ratio: \( \frac{1}{1/2} = 2 \), \( \frac{2}{1} = 2 \), \( \frac{4}{2} = 2 \).
   - This is a geometric sequence with a common ratio \( r = 2 \).

2. **Option b: \( 4, 9, 16, 25, \dots \)**
   - Ratio: \( \frac{9}{4} = 2.25 \), \( \frac{16}{9} \neq 2.25 \) (it's not constant).
   - This is not a geometric sequence.

3. **Option c: \( 5, 15, 45, 135, \dots \)**
   - Ratio: \( \frac{15}{5} = 3 \), \( \frac{45}{15} = 3 \), \( \frac{135}{45} = 3 \).
   - This is a geometric sequence with a common ratio \( r = 3 \).

4. **Option d: \( 1, -2, 4, -8, \dots \)**
   - Ratio: \( \frac{-2}{1} = -2 \), \( \frac{4}{-2} = -2 \), \( \frac{-8}{4} = -2 \).
   - This is a geometric sequence with a common ratio \( r = -2 \).

5. **Option e: \( 2, 0, -2, -4, -6, \dots \)**
   - The difference is not constant. It's an arithmetic sequence (not geometric).
   - This is not a geometric sequence.

6. **Option f: \( 8, 4, 0, -4, \dots \)**
   - The ratio is not defined because division by 0 is undefined.
   - This is not a geometric sequence.

7. **Option g: \( 2, -2, 2, -2, 2, \dots \)**
   - Ratio: \( \frac{-2}{2} = -1 \), \( \frac{2}{-2} = -1 \), \( \frac{-2}{2} = -1 \).
   - This is a geometric sequence with a common ratio \( r = -1 \).

8. **Option h: \( 1, \frac{1}{10}, \frac{1}{100}, \frac{1}{1000}, \dots \)**
   - Ratio: \( \frac{1/10}{1} = \frac{1}{10} \), \( \frac{1/100}{1/10} = \frac{1}{10} \), \( \frac{1/1000}{1/100} = \frac{1}{10} \).
   - This is a geometric sequence with a common ratio \( r = \frac{1}{10} \).

9. **Option i: \( 9, 6, 4, \frac{8}{3}, \dots \)**
   - Ratio: \( \frac{6}{9} = \frac{2}{3} \), \( \frac{4}{6} \neq \frac{2}{3} \) (it's not constant).
   - This is not a geometric sequence.

### Conclusion:
- The geometric sequences are:
  - **a.** Common ratio \( r = 2 \)
  - **c.** Common ratio \( r = 3 \)
  - **d.** Common ratio \( r = -2 \)
  - **g.** Common ratio \( r = -1 \)
  - **h.** Common ratio \( r = \frac{1}{10} \)

Would you like further clarification or details on any of the steps?

### Related questions:
1. How do you differentiate between arithmetic and geometric sequences?
2. What are some real-life examples of geometric sequences?
3. Can a geometric sequence have a common ratio of 0?
4. How does the formula for the nth term of a geometric sequence work?
5. What happens when the common ratio of a geometric sequence is a fraction?

### Tip:
In a geometric sequence, the sign and magnitude of the common ratio can greatly affect the behavior of the sequence—alternating signs if negative, shrinking if between 0 and 1, or growing rapidly if larger than 1.

Pada barisan berikut ini, manakah yang merupakan barisan geometri? Carilah rasionya!

Determine which sequences are geometric and calculate the common ratio. This problem covers key mathematical concepts such as geometric sequences and the common ratio. Suitable for Grades 10-12.

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