The sequence $4, 8, 16, 32$ is a geometric sequence with a common ratio of 2. To write a recursive formula, we describe each term $a_n$ in terms of the previous term $a_{n-1}$.

Recursive Formula:

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

Explanation:

• $a_1 = 4$: The first term is explicitly defined as 4.
• $a_n = 2 \cdot a_{n-1}$: Each term is obtained by multiplying the previous term by 2.

Would you like further details or examples on using this recursive formula?

Related Questions:

1. What is the explicit formula for this sequence?
2. How can the recursive formula be modified if the common ratio changes?
3. What happens if the first term changes but the ratio remains the same?
4. Can you derive the $n$-th term of the sequence directly from the recursive formula?
5. How does this sequence compare to an arithmetic sequence?

Tip:

In a geometric sequence, if you know the first term and the common ratio, you can always find any term by repeatedly applying the recursive formula.