Discover the Term-to-Term Rule: Analyzing the 3, 6, 9, 12 Sequence

Let's solve the problem step by step:

• Step 1: Identify the pattern in the sequence.

Looking at the sequence: 3, 6, 9, 12, ..., notice that each term is greater than the previous term by 3. This indicates a constant difference of 3.

• Step 2: Determine the term-to-term rule using this difference.

The sequence can be classified as an arithmetic sequence where the common difference $d$ is 3. In an arithmetic sequence, each term is given by the formula:

$a_n = a_1 + (n-1) \cdot d$

For this sequence, where $a_1 = 3$ and $d = 3$, we can rewrite the formula as:

$a_n = 3 + (n-1) \cdot 3$

Simplifying, we have:

$a_n = 3 + 3n - 3 = 3n$

• Step 3: Verify the term-to-term rule matches the sequence.

Check: When $n = 1$, $a_1 = 3 \cdot 1 = 3$.
When $n = 2$, $a_2 = 3 \cdot 2 = 6$.
And so on for the rest of the sequence.

Therefore, the rule $a_n = 3n$ correctly describes the sequence.

The answer choice $3n$ corresponds to choice number 2.

Therefore, the term-to-term rule for the sequence is $3n$.