The sequence given in the image is:

$7, 13, 19, 25, \ldots$

To find the general $n$-th term, we observe that the difference between consecutive terms is constant:

$13 - 7 = 6, \quad 19 - 13 = 6, \quad 25 - 19 = 6$

This tells us that the sequence is arithmetic, with a common difference of $d = 6$.

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

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

where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term of the sequence,
• $d$ is the common difference,
• $n$ is the position of the term.

Here, $a_1 = 7$ and $d = 6$, so the formula becomes:

$a_n = 7 + (n - 1) \cdot 6$

Simplifying:

$a_n = 7 + 6n - 6 = 6n + 1$

Thus, the formula for the $n$-th term is:

$a_n = 6n + 1$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How can you find the common difference in any arithmetic sequence?
2. What would be the 10th term of the sequence using the formula?
3. How does the formula for the sum of an arithmetic sequence relate to the $n$-th term?
4. Can you determine the $n$-th term formula for a geometric sequence?
5. What is the difference between arithmetic and geometric sequences?

Tip: Always verify if a sequence is arithmetic by checking if the difference between consecutive terms is constant.