To solve the problem and find the 9th term of the arithmetic sequence, we will analyze the given terms and calculate the common difference $d$ and apply the formula for the nth term:

The first few terms of the sequence are: $-5x + 2, -11x - 3, -17x - 8, \dots$

Step 1: Find the common difference $d$

The common difference $d$ in an arithmetic sequence is calculated as: $d = \text{second term} - \text{first term}$

$d = (-11x - 3) - (-5x + 2)$

Simplify: $d = -11x - 3 + 5x - 2$

$d = -6x - 5$

Step 2: General formula for the $n$th term

The general formula for the $n$th term of an arithmetic sequence is: $a_n = a_1 + (n-1) \cdot d$ where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Here: $a_1 = -5x + 2, \quad d = -6x - 5$

Step 3: Find the 9th term ($n = 9$)

Substitute into the formula: $a_9 = a_1 + (9-1) \cdot d$

$a_9 = (-5x + 2) + 8 \cdot (-6x - 5)$

Simplify: $a_9 = -5x + 2 - 48x - 40$

$a_9 = -53x - 38$

Final Answer:

The 9th term of the sequence is: $\boxed{-53x - 38}$

Would you like more details or have further questions? Here are five related questions for exploration:

1. What is the general formula for any term in this sequence?
2. How can we verify that the common difference is consistent across the sequence?
3. How would the formula change if the first term were different?
4. What happens if $x$ has a specific value, e.g., $x = 2$?
5. How can we identify whether a sequence is arithmetic or not?

Tip: Always verify your calculated common difference $d$ by checking multiple consecutive terms to ensure consistency.