https://file.aiquickdraw.com/mathbot/file/17343608145961yexecmn.jpg

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

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### **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 \( r = \frac{1}{3} \):  
   \[
   a_n = 540 \cdot \left(\frac{1}{3}\right)^{n-1}.
   \]

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### **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**: \( a_n = 540 \cdot \left(\frac{1}{3}\right)^{n-1} \)

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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.

Write a recursive formula and an explicit formula for each geometric sequence: 
1. 9, 18, 36, 72, ...
2. 540, 180, 60, 20, ...

Discover how to write recursive and explicit formulas for geometric sequences like 9, 18, 36, 72 and 540, 180, 60, 20. Learn step-by-step solutions, including identifying the common ratio and applying formulas. Ideal for students in Grades 8-10.

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