To find the recursive formula for the sequence $a_n$ where the sequence is $8, 48, 288, 1728, \ldots$, we start by identifying the pattern of the sequence.

Step 1: Identifying the Pattern

Let’s list the first few terms:

• $a_1 = 8$
• $a_2 = 48$
• $a_3 = 288$
• $a_4 = 1728$

Step 2: Finding the Recursive Relationship

To find the recursive formula, calculate the ratio between consecutive terms:

$\frac{a_2}{a_1} = \frac{48}{8} = 6$

$\frac{a_3}{a_2} = \frac{288}{48} = 6$

$\frac{a_4}{a_3} = \frac{1728}{288} = 6$

It appears that each term is multiplied by 6 to get the next term.

Step 3: Writing the Recursive Formula

Given the relationship between terms, the recursive formula is:

$a_n = 6 \cdot a_{n-1} \quad \text{for } n \geq 2$

with the initial term:

$a_1 = 8$

Final Recursive Formula

$\boxed{a_n = 6 \cdot a_{n-1}, \quad a_1 = 8}$

This formula correctly generates the sequence $8, 48, 288, 1728, \ldots$.

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Would you like further details, or do you have any questions? Here are some related questions:

1. How would you write the explicit formula for this sequence?
2. What is the 10th term of this sequence using the recursive formula?
3. Can you find the sum of the first 5 terms of the sequence?
4. How does this sequence grow compared to an arithmetic sequence?
5. What would the sequence look like if the ratio changed from 6 to another number?

Tip: Always check the first few terms with your recursive formula to confirm its correctness!