The sequence in the image is $3, 12, 21, 30, \dots$. This is an arithmetic sequence where each term increases by the same value. Let's break down how to find the rule for the $n$-th term.

1. Find the common difference:

   • The difference between consecutive terms is $12 - 3 = 9$, $21 - 12 = 9$, and $30 - 21 = 9$. So, the common difference $d = 9$.

2. Find the first term:

   • The first term of the sequence is $a_1 = 3$.

3. Use the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n - 1) \cdot d$ Substituting $a_1 = 3$ and $d = 9$: $a_n = 3 + (n - 1) \cdot 9$ Simplifying: $a_n = 3 + 9n - 9 = 9n - 6$

Thus, the $n$-th term rule for this sequence is $9n - 6$.

Would you like more details or clarifications on any of these steps?

Here are 5 related questions to expand your understanding:

1. What is the general formula for the $n$-th term of an arithmetic sequence?
2. How would the rule change if the first term was different?
3. Can you find the 10th term using the formula $9n - 6$?
4. What if the common difference was negative? How would the sequence behave?
5. How do you determine if a sequence is arithmetic or geometric?

Tip: Always check the first few differences between terms to identify if a sequence is arithmetic (constant difference) or geometric (constant ratio).