Find the sum of the first 8 terms of the following sequence. Round to the nearest hundredth if necessary.
4, comma, 6, comma, 9, comma, point, point, .
4,6,9,...

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.

Find the sum of the first 8 terms of the following sequence. Round to the nearest hundredth if necessary. 4, 6, 9, ...

Discover how to calculate the sum of the first 8 terms in a geometric sequence (4, 6, 9, ...) using the geometric series sum formula. This detailed solution explains key concepts, such as common ratio and sum formulas, and is perfect for students in Grades 9-12.

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