The first four terms of the arithmetic sequence are given as: $-11, -5, 1, 7$. To find the equation for the $n$-th term of the sequence $a_n$, we need to determine the common difference $d$ and the first term $a_1$.

1. Find the common difference $d$:

The common difference is the difference between any two consecutive terms. Let's find $d$:

$d = -5 - (-11) = -5 + 11 = 6$

Thus, the common difference $d = 6$.

2. General formula for the $n$-th term of an arithmetic sequence:

The formula for the $n$-th term of an arithmetic sequence is given by:

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

Where:

• $a_1 = -11$ (the first term)
• $d = 6$ (the common difference)

3. Substitute $a_1$ and $d$ into the formula:

$a_n = -11 + (n - 1) \cdot 6$

Simplifying:

$a_n = -11 + 6n - 6$

$a_n = 6n - 17$

Thus, the equation for the $n$-th term of the arithmetic sequence is:

$a_n = 6n - 17$

Would you like more details or have any questions?

---

Here are 5 related questions to further your understanding:

1. How do you derive the formula for the sum of an arithmetic sequence?
2. What is the sum of the first 10 terms of this arithmetic sequence?
3. Can you find the 20th term of this arithmetic sequence using the formula $a_n = 6n - 17$?
4. What happens to the sequence if the common difference $d$ is negative?
5. How does an arithmetic sequence differ from a geometric sequence?

Tip: The common difference $d$ is always the same in an arithmetic sequence, no matter which two consecutive terms you choose!