To solve this problem, we first need to identify the type of sequence and use the relevant formulas.

Step 1: Determine the Sequence Type

The given sequence is: $3, 6, 9, \dots$ This is an arithmetic sequence where each term increases by a common difference. Here, the first term $a = 3$ and the common difference $d = 6 - 3 = 3$.

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

The general formula for the $n$-th term of an arithmetic sequence is: $a_n = a + (n - 1)d$ We are given that $a_8 = 24$, so let’s plug in the values to verify or further calculate any unknowns:

$a_8 = 3 + (8 - 1) \cdot 3$ $a_8 = 3 + 7 \cdot 3$ $a_8 = 3 + 21 = 24$ This matches, confirming that our values for $a$ and $d$ are correct.

Step 3: Find the Sum of the First 8 Terms

The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} (2a + (n - 1)d)$ For the first 8 terms ($n = 8$): $S_8 = \frac{8}{2} (2 \cdot 3 + (8 - 1) \cdot 3)$ $S_8 = 4 (6 + 21)$ $S_8 = 4 \cdot 27 = 108$

Answer

The sum of the first 8 terms of the series is: $\boxed{108}$

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

Here are some related questions to explore further:

1. How would the sum change if the common difference were different?
2. What if the sequence started with a different first term?
3. Can you find the sum of the first 20 terms of this sequence?
4. How would this problem change if it were a geometric sequence?
5. How do you derive the formula for the sum of an arithmetic sequence?

Tip: When working with sequences, identifying the type (arithmetic or geometric) is crucial as it determines the approach for finding sums and terms.