Identify the Term-to-Term Rule in the Arithmetic Sequence: 3, 7, 11, 15

To solve this problem, we'll follow these steps:

• Identify the common difference.
• Use the arithmetic sequence formula to find a general rule.
• Compare the rule with the given choices to identify the correct one.

Now, let's work through each step:
Step 1: Identify the common difference.
By examining the sequence: $3, 7, 11, 15, \ldots$, we find that the difference ($d$) between each pair of consecutive terms is $7 - 3 = 4$, $11 - 7 = 4$, and $15 - 11 = 4$. Hence, the common difference $d$ is 4.
Step 2: Use the arithmetic sequence formula.
The first term $a_1$ is 3. Using the formula for the nth term of an arithmetic sequence: $a_n = a_1 + (n-1)d$
Substitute $a_1 = 3$ and $d = 4$:
$a_n = 3 + (n-1) \cdot 4$
Simplify this equation:
$a_n = 3 + 4n - 4$
$a_n = 4n - 1$
Step 3: Compare the rule with the provided choices.
The derived formula is $a_n = 4n - 1$ which matches the choice $\textbf{(2)}: 4n - 1$.

Therefore, the term-to-term rule of the sequence is $\mathbf{4n - 1}$.