To solve this, we need both recursive and explicit formulas for each given geometric sequence. Let's tackle them one at a time.

Problem 4: Sequence: 9, 18, 36, 72, ...

1. Identify the common ratio:
   The common ratio $r$ is obtained by dividing any term by the previous one:
   $r = \frac{\text{next term}}{\text{previous term}} = \frac{18}{9} = 2.$ So the sequence multiplies by $2$ at each step.

2. Recursive formula:
   The recursive formula defines each term based on the previous one:
   $a_n = a_{n-1} \cdot r, \quad \text{where } a_1 = 9 \text{ and } r = 2.$ Therefore:
   $a_n = a_{n-1} \cdot 2 \quad \text{with } a_1 = 9.$

3. Explicit formula:
   The explicit formula for a geometric sequence is:
   $a_n = a_1 \cdot r^{n-1}.$ Substituting $a_1 = 9$ and $r = 2$:
   $a_n = 9 \cdot 2^{n-1}.$

Problem 5: Sequence: 540, 180, 60, 20, ...

1. Identify the common ratio:
   The common ratio $r$ is obtained by dividing any term by the previous one:
   $r = \frac{\text{next term}}{\text{previous term}} = \frac{180}{540} = \frac{1}{3}.$ So the sequence multiplies by $\frac{1}{3}$ at each step.

2. Recursive formula:
   The recursive formula defines each term based on the previous one:
   $a_n = a_{n-1} \cdot r, \quad \text{where } a_1 = 540 \text{ and } r = \frac{1}{3}.$ Therefore:
   $a_n = a_{n-1} \cdot \frac{1}{3} \quad \text{with } a_1 = 540.$

3. Explicit formula:
   The explicit formula for a geometric sequence is:
   $a_n = a_1 \cdot r^{n-1}.$ Substituting $a_1 = 540$ and $r = \frac{1}{3}$:
   $a_n = 540 \cdot \left(\frac{1}{3}\right)^{n-1}.$

Final Answers:

For sequence $9, 18, 36, 72, \dots$:

• Recursive: $a_n = a_{n-1} \cdot 2, \, a_1 = 9$
• Explicit: $a_n = 9 \cdot 2^{n-1}$

For sequence $540, 180, 60, 20, \dots$:

• Recursive: $a_n = a_{n-1} \cdot \frac{1}{3}, \, a_1 = 540$
• Explicit: $a_n = 540 \cdot \left(\frac{1}{3}\right)^{n-1}$

Do you need any further explanation or examples? Let me know!

Related Questions:

1. What is the general form of a geometric sequence's explicit formula?
2. How do you determine the common ratio in a geometric sequence?
3. How do recursive and explicit formulas differ in application?
4. Can a geometric sequence have a negative common ratio? Give an example.
5. How do you solve for a specific term in a geometric sequence using its explicit formula?

Tip:

To quickly check if a sequence is geometric, divide consecutive terms to ensure the ratio is constant.