The given sequence is: 4, 6, 9, ...

We can see that this is an arithmetic sequence, where the difference between consecutive terms is constant. To verify, let's check the common difference $d$: $d = 6 - 4 = 2$ $d = 9 - 6 = 3$ Since the difference is not constant, this is not an arithmetic sequence. Next, let's check if it is a geometric sequence by finding the ratio between consecutive terms: $\frac{6}{4} = 1.5$ $\frac{9}{6} = 1.5$ This confirms that the common ratio $r = 1.5$, so it is a geometric sequence.

Formula for the sum of the first $n$ terms of a geometric sequence:

$S_n = a \frac{1 - r^n}{1 - r}$ Where:

• $a = 4$ (the first term)
• $r = 1.5$ (the common ratio)
• $n = 8$ (the number of terms)

Substituting into the formula:

$S_8 = 4 \times \frac{1 - (1.5)^8}{1 - 1.5}$ Now let's calculate it:

$1.5^8 = 25.62890625$ $S_8 = 4 \times \frac{1 - 25.62890625}{1 - 1.5}$ $S_8 = 4 \times \frac{-24.62890625}{-0.5}$ $S_8 = 4 \times 49.2578125 = 197.03125$

Therefore, the sum of the first 8 terms is:

$S_8 \approx 197.03$

Do you want any details or have any questions?

Here are 5 related questions to explore further:

1. What is the formula for the sum of an arithmetic sequence?
2. How can you find the nth term of a geometric sequence?
3. How do geometric sequences differ from arithmetic sequences?
4. What happens if the common ratio $r$ is negative in a geometric series?
5. How do you solve for the sum of an infinite geometric series?

Tip: Always check if a sequence is arithmetic or geometric by comparing differences or ratios between terms.