The given pattern is an arithmetic sequence: 6, 12, 18, 24, ...

In this arithmetic sequence:

• The first term ($a$) is 6.
• The common difference ($d$) is the difference between consecutive terms, which is $12 - 6 = 6$.

Relation for the Pattern Rule:

The general formula for the $n$-th term of an arithmetic sequence is given by: $T_n = a + (n - 1) \cdot d$ where:

• $T_n$ is the $n$-th term,
• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Substituting the values: $T_n = 6 + (n - 1) \cdot 6$ Simplify the expression: $T_n = 6 + 6n - 6$ $T_n = 6n$

Finding the 12th term:

To find the 12th term ($T_{12}$): $T_{12} = 6 \cdot 12 = 72$

Therefore, the 12th term is 72.

Would you like more details or have any other questions?

Here are five related questions for further exploration:

1. How would the formula change if the common difference was not constant?
2. What is the sum of the first 12 terms in this sequence?
3. How can you determine the common difference in a sequence if it isn't given?
4. Can you create a recursive formula for this sequence?
5. How does the formula for the $n$-th term change for a geometric sequence?

Tip: Arithmetic sequences always have a constant difference between consecutive terms.