The image shows a worksheet on recursive formulas for geometric sequences. Let me help solve the given problems:

Problem 1: $4, 8, 16, 32, \dots$

This is a geometric sequence where each term is multiplied by 2 to get the next term.

Recursive formula: $a_1 = 4, \quad a_n = a_{n-1} \cdot 2 \quad \text{for } n \geq 2$

Problem 3: $3, 6, 12, 24, \dots$

This is another geometric sequence, where each term is multiplied by 2 to get the next term.

Recursive formula: $a_1 = 3, \quad a_n = a_{n-1} \cdot 2 \quad \text{for } n \geq 2$

Would you like me to assist with the other questions or provide detailed explanations for these?

Expanding Questions

1. How would you derive a recursive formula for an arithmetic sequence instead of a geometric one?
2. What is the difference between recursive and explicit formulas for sequences?
3. Can you convert these recursive formulas into explicit formulas?
4. How do you determine the ratio in a geometric sequence if not given explicitly?
5. What happens to the recursive formula when the common ratio is a fraction?

Tip

When identifying recursive formulas for geometric sequences, always find the ratio by dividing any term by its previous term!