The arithmetic sequence presented in the image is:

$-4x + 5, -10x + 2, -16x - 1, \ldots$

We are tasked with finding the 12th term of this sequence.

Step 1: Identify the First Term ($a_1$)

The first term of the sequence is:

$a_1 = -4x + 5$

Step 2: Find the Common Difference ($d$)

The common difference $d$ in an arithmetic sequence is the difference between any two consecutive terms.

Let's subtract the first term from the second term:

$d = (-10x + 2) - (-4x + 5) = -10x + 2 + 4x - 5 = -6x - 3$

So, the common difference $d = -6x - 3$.

Step 3: Use the Formula for the $n$-th Term

The general formula for the $n$-th term $a_n$ of an arithmetic sequence is:

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

To find the 12th term $a_{12}$, substitute $n = 12$, $a_1 = -4x + 5$, and $d = -6x - 3$ into the formula:

$a_{12} = (-4x + 5) + (12 - 1) \cdot (-6x - 3)$ $a_{12} = (-4x + 5) + 11 \cdot (-6x - 3)$ $a_{12} = (-4x + 5) + (-66x - 33)$ $a_{12} = -4x + 5 - 66x - 33$ $a_{12} = -70x - 28$

Final Answer:

The 12th term of the arithmetic sequence is:

$a_{12} = -70x - 28$

Would you like more details on any step, or do you have any questions?

Here are some related questions to consider:

1. What is the general term of this arithmetic sequence?
2. How do you find the sum of the first 12 terms of the sequence?
3. What is the 20th term of this arithmetic sequence?
4. How would the result change if $x$ is a specific number (e.g., $x = 2$)?
5. Can you derive the formula for any term of an arithmetic sequence?

Tip: In an arithmetic sequence, knowing the first term and common difference lets you calculate any term using a simple formula!