The sequence provided is an arithmetic sequence: $-4, -1, 2, 5, \dots$

Step 1: Identify the common difference

In an arithmetic sequence, the common difference ($d$) is constant between consecutive terms.

$d = -1 - (-4) = 3$ $d = 2 - (-1) = 3$ $d = 5 - 2 = 3$

Thus, the common difference $d = 3$.

Step 2: Find the first term

The first term of the sequence is $a_1 = -4$.

Step 3: 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:

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

Substitute $a_1 = -4$ and $d = 3$:

$a_n = -4 + (n - 1) \cdot 3$ $a_n = -4 + 3n - 3$ $a_n = 3n - 7$

Conclusion:

The $n$th term rule for the sequence is:

$a_n = 3n - 7$

Would you like more details or have any questions? Here are some related questions to explore:

1. How do you find the sum of the first $n$ terms in an arithmetic sequence?
2. What happens to the sequence if the common difference changes?
3. How can you determine the number of terms in a finite arithmetic sequence?
4. How does the arithmetic sequence differ from a geometric sequence?
5. Can you derive the nth term formula for a sequence given any two terms?

Tip: Always verify the common difference in each step to ensure the sequence remains arithmetic.