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.