To solve this problem, we need to determine which formula represents the $n$th term of the sequence $\{2, -1, -4, -7, \dots\}$.

Step 1: Observe the sequence

The sequence given is: $2, -1, -4, -7, \dots$

To identify the formula, calculate the difference between consecutive terms:

• $-1 - 2 = -3$
• $-4 - (-1) = -3$
• $-7 - (-4) = -3$

The common difference is $-3$, meaning the sequence is arithmetic.

Step 2: General form of an arithmetic sequence

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

• $a_1$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Step 3: Substitute values

Here:

• $a_1 = 2$,
• $d = -3$.

Substitute into the formula: $a_n = 2 + (n - 1)(-3)$ Simplify: $a_n = 2 - 3(n - 1)$ $a_n = 2 - 3n + 3$ $a_n = -3n + 5$

Step 4: Verify the formula

Now, test the formula $a_n = -3n + 5$ with the terms of the sequence:

• For $n = 1$: $a_1 = -3(1) + 5 = 2$ (correct),
• For $n = 2$: $a_2 = -3(2) + 5 = -1$ (correct),
• For $n = 3$: $a_3 = -3(3) + 5 = -4$ (correct),
• For $n = 4$: $a_4 = -3(4) + 5 = -7$ (correct).

Final Answer:

The correct equation is: $\boxed{D. \, a_n = -3n + 5}$

Do you have any questions or need further clarification?

Here are 5 related questions to consider:

1. How do we derive the general formula for an arithmetic sequence?
2. What distinguishes an arithmetic sequence from other types of sequences?
3. Can the common difference in an arithmetic sequence be a fraction?
4. How would the formula change if the sequence started with a negative term?
5. How can we find the sum of the first $n$ terms of this arithmetic sequence?

Tip: Always verify the formula by testing multiple terms in the sequence to