To find the $n$-th term rule for this arithmetic sequence (6, 8, 10, 12, ...), we can apply the standard formula for an arithmetic sequence:

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

Where:

• $a_n$ is the $n$-th term.
• $a_1$ is the first term.
• $d$ is the common difference.

Step 1: Identify the first term ($a_1$)

From the sequence, the first term $a_1 = 6$.

Step 2: Determine the common difference ($d$)

The difference between consecutive terms is: $d = 8 - 6 = 2$

Step 3: Write the formula

Substitute the values of $a_1 = 6$ and $d = 2$ into the formula:

$a_n = 6 + (n - 1) \cdot 2$

Simplify:

$a_n = 6 + 2n - 2$ $a_n = 2n + 4$

Thus, the $n$-th term rule for the sequence is:

$\boxed{a_n = 2n + 4}$

Would you like further details or have any questions?

Here are some related questions to consider:

1. What is the 10th term of this sequence?
2. How would the formula change if the common difference were negative?
3. What if the first term of the sequence was 5 instead of 6?
4. Can you derive the sum of the first $n$ terms of this sequence?
5. How would you generalize this to sequences with fractional differences?

Tip: In arithmetic sequences, knowing the common difference $d$ allows you to easily generate any term by using the formula!