To find the 12th term of the given arithmetic sequence, we first need to determine the common difference and then use the formula for the $n$th term of an arithmetic sequence.

Step 1: Identify the first term ($a_1$) and the common difference ($d$)

The given sequence is: $5x - 6, \quad 9x - 10, \quad 13x - 14, \ldots$

• The first term $a_1 = 5x - 6$.
• The second term $a_2 = 9x - 10$.

The common difference $d$ is calculated as: $d = a_2 - a_1 = (9x - 10) - (5x - 6)$ $d = 9x - 10 - 5x + 6 = 4x - 4$

Step 2: Use the formula for the $n$th term of an arithmetic sequence

The formula for the $n$th term ($a_n$) of an arithmetic sequence is: $a_n = a_1 + (n-1) \cdot d$

Step 3: Substitute the values for $n = 12$, $a_1$, and $d$

$a_{12} = (5x - 6) + (12 - 1) \cdot (4x - 4)$ $a_{12} = (5x - 6) + 11 \cdot (4x - 4)$ $a_{12} = (5x - 6) + (44x - 44)$ $a_{12} = 5x - 6 + 44x - 44$ $a_{12} = 49x - 50$

So, the 12th term of the arithmetic sequence is $\boxed{49x - 50}$.

Would you like to go over the details, or do you have any further questions? Here are some related questions you might find interesting:

1. How do you find the sum of the first 12 terms of this arithmetic sequence?
2. What is the 20th term of this sequence?
3. How do you determine whether a sequence is arithmetic?
4. Can you find the common difference of an arithmetic sequence from any two terms?
5. How would you solve for $x$ if the 12th term is given?

Tip: In arithmetic sequences, knowing just two terms is enough to find the common difference and thus determine any term in the sequence.