To solve this problem, we need to use the general formula for the $n$th term of an arithmetic sequence:

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

where:

• $a_n$ is the $n$th term,
• $a_1$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Given:

• The common difference $d = -6$,
• The 5th term $a_5 = -34$.

Step 1: Find the first term $a_1$

Using the formula for the 5th term: $a_5 = a_1 + (5 - 1) \cdot d$ Substitute the given values: $-34 = a_1 + 4 \cdot (-6)$ $-34 = a_1 - 24$ $a_1 = -34 + 24 = -10$

Step 2: Find the 30th term $a_{30}$

Now, use the first term and the common difference to find the 30th term: $a_{30} = a_1 + (30 - 1) \cdot d$ $a_{30} = -10 + 29 \cdot (-6)$ $a_{30} = -10 - 174 = -184$

So, the 30th term of the arithmetic sequence is $-184$.

Would you like more details on this solution, or do you have any other questions?

Here are five related questions you might find interesting:

1. How do you derive the formula for the $n$th term of an arithmetic sequence?
2. If the 15th term of the sequence were known, how would that change the calculation?
3. What would be the sum of the first 30 terms of this arithmetic sequence?
4. How do arithmetic sequences differ from geometric sequences?
5. What happens to the sequence if the common difference is positive instead of negative?

Tip: Always ensure the sequence's first term and common difference are accurately identified before solving for any term or sum in an arithmetic sequence.