The given sequence is:
$3, 7, 11, 15, \dots$

We observe that each term increases by 4, which suggests the sequence is an arithmetic sequence where the common difference $d = 4$.

Step 1: Find the recursive formula

For an arithmetic sequence, the recursive formula can be written as:

$a_n = a_{n-1} + d$ where $a_n$ is the nth term, $a_{n-1}$ is the previous term, and $d$ is the common difference.

Since the common difference is $d = 4$, the recursive formula becomes:

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

Step 2: Define the first term

We also need to define the first term explicitly. From the given sequence, the first term $a_1 = 3$.

Thus, the full recursive formula for the sequence is:

$a_1 = 3$ $a_n = a_{n-1} + 4 \quad \text{for} \quad n \geq 2$

Final Recursive Formula:

$a_1 = 3$ $a_n = a_{n-1} + 4 \quad \text{for} \quad n \geq 2$

Would you like further details or an explanation of any part of the solution? Here are some related questions to explore:

1. How do you identify whether a sequence is arithmetic or geometric?
2. Can you find the nth term formula for an arithmetic sequence?
3. What are the general characteristics of recursive formulas for arithmetic sequences?
4. How do you extend recursive formulas to other types of sequences, like geometric ones?
5. What happens if the common difference is negative or zero in an arithmetic sequence?

Tip: When working with recursive sequences, always identify the common difference (in arithmetic sequences) or ratio (in geometric sequences) first!