We are given the first four terms of an arithmetic sequence: $-11, -5, 1, 7$. The goal is to determine the equation for $a_n$, the general term of the sequence.

Step 1: Identify the Common Difference

To find the common difference, we subtract any term from the previous term: $d = -5 - (-11) = 6$ $d = 1 - (-5) = 6$ $d = 7 - 1 = 6$ The common difference $d = 6$.

Step 2: Use the General Formula for an Arithmetic Sequence

The general form 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.

From the problem, $a_1 = -11$ and $d = 6$. Substituting these values into the formula: $a_n = -11 + (n - 1) \cdot 6$

Step 3: Simplify the Equation

$a_n = -11 + 6(n - 1)$ $a_n = -11 + 6n - 6$ $a_n = 6n - 17$

Step 4: Match with Given Options

Comparing the simplified equation $a_n = 6n - 17$ with the given options, we find that it corresponds to option B: $B. \ a_n = 6(n - 1) - 11$

Final Answer: Option B is correct.

Would you like further details or clarifications?

Here are five questions that build upon this concept:

1. How do we derive the formula for the general term of an arithmetic sequence?
2. What are the key characteristics of an arithmetic sequence?
3. How would the formula change if the common difference were negative?
4. Can the first term $a_1$ be any value, or does it follow specific rules?
5. How can you determine the number of terms in an arithmetic sequence given a particular term?

Tip: When solving arithmetic sequence problems, always ensure to clearly identify both the first term and the common difference before attempting to derive the general term.