To solve this problem, we need to find the $n$-th term formula for the given arithmetic sequence and then calculate $a_{30}$.

Step 1: Identify the Pattern of the Sequence

The sequence given is: $6, 12, 18, 24, \ldots$

This is an arithmetic sequence because each term increases by the same amount. The common difference $d$ can be calculated as: $d = 12 - 6 = 6$

The first term $a_1$ is: $a_1 = 6$

Step 2: Write the Formula for the $n$-th Term

The general formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n - 1)d$

Substitute $a_1 = 6$ and $d = 6$ into the formula: $a_n = 6 + (n - 1) \cdot 6$ $a_n = 6 + 6n - 6$ $a_n = 6n$

Thus, the equation for the $n$-th term is: $a_n = 6n$

Step 3: Find $a_{30}$

Now, substitute $n = 30$ into the formula: $a_{30} = 6 \cdot 30 = 180$

Answer

The $30$-th term of the sequence, $a_{30}$, is: $a_{30} = 180$

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. How can we find the sum of the first 30 terms of this sequence?
2. What is the 50th term of this arithmetic sequence?
3. How does the formula for the $n$-th term of an arithmetic sequence change if we know only the common difference?
4. How can we verify that this sequence is indeed arithmetic?
5. How would the sequence change if each term increased by a different value instead of a constant difference?

Tip: For arithmetic sequences, the $n$-th term formula is straightforward to derive once you know the first term and the common difference.