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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!

What is the nth term rule for the arithmetic sequence: 6, 8, 10, 12, ...?

Discover how to find the nth term rule for the arithmetic sequence 6, 8, 10, 12. This solution explains arithmetic sequences, common differences, and algebraic formulas for middle school math students.

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