https://file.aiquickdraw.com/mathbot/file/17266554916765em0s67i.jpg

Let's solve both geometric series problems from the image.

### Formula for the sum of a finite geometric series:
\[
S_n = \frac{a_1 (1 - r^n)}{1 - r}
\]
where:
- \(S_n\) is the sum of the first \(n\) terms,
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the number of terms.

### Problem 1:
**Find the sum of the first six terms of the geometric sequence: 3, 6, 12, 24, ...**

**Step 1:** Identify the variables.
- \(a_1 = 3\) (the first term),
- \(r = \frac{6}{3} = 2\) (the common ratio),
- \(n = 6\) (the number of terms).

**Step 2:** Use the formula.
\[
S_6 = \frac{3(1 - 2^6)}{1 - 2}
\]
\[
S_6 = \frac{3(1 - 64)}{-1}
\]
\[
S_6 = \frac{3(-63)}{-1} = 189
\]
So, the sum of the first six terms is **189**.

---

### Problem 2:
**Find the sum of 10 terms of the sequence: 1, \(\frac{1}{2}\), \(\frac{1}{4}\), ...**

**Step 1:** Identify the variables.
- \(a_1 = 1\) (the first term),
- \(r = \frac{\frac{1}{2}}{1} = \frac{1}{2}\) (the common ratio),
- \(n = 10\) (the number of terms).

**Step 2:** Use the formula.
\[
S_{10} = \frac{1(1 - \left(\frac{1}{2}\right)^{10})}{1 - \frac{1}{2}}
\]
\[
S_{10} = \frac{1(1 - \frac{1}{1024})}{\frac{1}{2}}
\]
\[
S_{10} = \frac{1(1 - 0.0009765625)}{0.5} = \frac{0.9990234375}{0.5}
\]
\[
S_{10} = 1.998046875
\]
So, the sum of the first 10 terms is approximately **1.998**.

---

Do you need any further clarifications or more details on these steps?

Here are 5 questions to expand on this:
1. What happens if the common ratio \(r\) is greater than 1 for an infinite geometric series?
2. How does the sum of a geometric series change if the common ratio is negative?
3. How can you calculate the sum of an infinite geometric series when \( |r| < 1 \)?
4. How is the geometric series applied in real-world financial models?
5. Can you think of an example where a geometric sequence models exponential growth?

**Tip:** For an infinite geometric series where \( |r| < 1 \), the sum formula simplifies to \( \frac{a_1}{1 - r} \).

Find the sum of the first six terms of the geometric sequence 3, 6, 12, 24,... and Find the sum of 10 terms of the sequence: 1, 1/2, 1/4,...

Learn how to find the sum of a finite geometric sequence with examples like 3, 6, 12, 24,... and 1, 1/2, 1/4,... Explore the formula for the sum of geometric series and see step-by-step solutions suitable for students in Grades 9-12.

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