The formula for the $n$th term of the sequence $3, 6, 9, 12$ is given by:

To determine the $n$th term of a sequence, it is essential to identify a pattern or rule governing the sequence. In this case, the sequence $3, 6, 9, 12$ is classified as an arithmetic sequence, where each term increases by a consistent amount. The difference between consecutive terms is $3$ (for example, $6 - 3 = 3$, $9 - 6 = 3$, and so forth). This consistent difference is referred to as the common difference.

To derive the $n$th term formula, we start by recognizing that the first term (when $n = 1$) is $3$. The second term (when $n = 2$) is $6$, which can be expressed as $3 \times 2$. Similarly, the third term (when $n = 3$) is $9$, or $3 \times 3$. This pattern continues, demonstrating that each term is $3$ multiplied by its position in the sequence, denoted by $n$. Thus, the $n$th term can be succinctly expressed as:

Using this formula, you can easily calculate any term in the sequence. For instance, to find the 5th term, substitute $n = 5$ into the formula:

Therefore, the 5th term is $15$. This method is particularly advantageous for quickly determining terms without the need to list all preceding terms, especially for larger values of $n$.