To determine the nth-term rule for the given arithmetic sequence, follow these steps:

Step 1: Identify the common difference

The sequence is: 3, 12, 21, 30.
The difference between consecutive terms is:

• $12 - 3 = 9$
• $21 - 12 = 9$
• $30 - 21 = 9$

So, the common difference ($d$) is $9$.

Step 2: Formulate the nth-term formula

The general formula for the nth term of an arithmetic sequence is:
$T_n = a + (n-1) \cdot d$
where:

• $a$ is the first term ($3$),
• $d$ is the common difference ($9$).

Substituting $a = 3$ and $d = 9$ into the formula:
$T_n = 3 + (n-1) \cdot 9$

Step 3: Simplify the formula

Expand and simplify:
$T_n = 3 + 9n - 9$
$T_n = 9n - 6$

Final Answer:

The nth-term rule is:
$T_n = 9n - 6$

Let me know if you want further explanation!

Related Questions:

1. What is the 10th term of this sequence using the nth-term formula?
2. How can you verify the nth-term formula with the given terms in the sequence?
3. What is the sum of the first 20 terms of this sequence?
4. Can this formula be used for negative values of $n$? Why or why not?
5. How would you derive the nth-term formula for a geometric sequence instead?

Tip:

To check if your nth-term formula is correct, substitute $n = 1, 2, 3, \ldots$ and ensure the results match the sequence values.