I. Find the sum of the following geometric sequence:
1. 1, 2, 4, 8, ..., A12
2. 3, 6, 12, 24, ..., A8
3. 512, 256, 128, ..., A6
4. 12, 18, 27, ..., A9
5. 2, 10, 50, ..., A6

To find the sum of each geometric sequence, we use the sum formula for a geometric sequence:

\[
S_n = a \cdot \frac{1 - r^n}{1 - r} \quad \text{if} \quad |r| < 1
\]
or
\[
S_n = a \cdot \frac{r^n - 1}{r - 1} \quad \text{if} \quad r > 1
\]
where:
- \(S_n\) is the sum of the first \(n\) terms,
- \(a\) is the first term of the sequence,
- \(r\) is the common ratio, and
- \(n\) is the number of terms.

### 1. Sequence: 1, 2, 4, 8, ..., A12
- First term: \(a = 1\)
- Common ratio: \(r = 2\)
- Number of terms: \(n = 12\)

The sum is:

\[
S_{12} = 1 \cdot \frac{2^{12} - 1}{2 - 1} = 2^{12} - 1 = 4096 - 1 = 4095
\]

### 2. Sequence: 3, 6, 12, 24, ..., A8
- First term: \(a = 3\)
- Common ratio: \(r = 2\)
- Number of terms: \(n = 8\)

The sum is:

\[
S_8 = 3 \cdot \frac{2^8 - 1}{2 - 1} = 3 \cdot (256 - 1) = 3 \cdot 255 = 765
\]

### 3. Sequence: 512, 256, 128, ..., A6
- First term: \(a = 512\)
- Common ratio: \(r = \frac{1}{2}\)
- Number of terms: \(n = 6\)

The sum is:

\[
S_6 = 512 \cdot \frac{1 - \left(\frac{1}{2}\right)^6}{1 - \frac{1}{2}} = 512 \cdot \frac{1 - \frac{1}{64}}{\frac{1}{2}} = 512 \cdot \frac{\frac{63}{64}}{\frac{1}{2}} = 512 \cdot \frac{63}{32} = 1008
\]

### 4. Sequence: 12, 18, 27, ..., A9
- First term: \(a = 12\)
- Common ratio: \(r = \frac{3}{2}\)
- Number of terms: \(n = 9\)

The sum is:

\[
S_9 = 12 \cdot \frac{\left(\frac{3}{2}\right)^9 - 1}{\frac{3}{2} - 1} = 12 \cdot \frac{(1.5)^9 - 1}{0.5}
\]
\[
S_9 = 12 \cdot \frac{38.443359375 - 1}{0.5} = 12 \cdot \frac{37.443359375}{0.5} = 12 \cdot 74.88671875 = 898.640625
\]

### 5. Sequence: 2, 10, 50, ..., A6
- First term: \(a = 2\)
- Common ratio: \(r = 5\)
- Number of terms: \(n = 6\)

The sum is:

\[
S_6 = 2 \cdot \frac{5^6 - 1}{5 - 1} = 2 \cdot \frac{15625 - 1}{4} = 2 \cdot \frac{15624}{4} = 2 \cdot 3906 = 7812
\]

### Summary:
1. \( S_{12} = 4095 \)
2. \( S_8 = 765 \)
3. \( S_6 = 1008 \)
4. \( S_9 \approx 898.64 \)
5. \( S_6 = 7812 \)

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How does changing the common ratio affect the sum of a geometric series?
2. Can the sum of a geometric series be infinite?
3. How does the sum change if the first term is negative?
4. What is the difference between a geometric sequence and an arithmetic sequence?
5. How do we find the sum of a geometric series with a common ratio less than 1?

**Tip**: When the common ratio is a fraction between 0 and 1, the terms of the geometric sequence decrease rapidly, leading to a smaller sum.

Find the sum of the following geometric sequences: 1. 1, 2, 4, 8, ..., A12; 2. 3, 6, 12, 24, ..., A8; 3. 512, 256, 128, ..., A6; 4. 12, 18, 27, ..., A9; 5. 2, 10, 50, ..., A6.

Explore detailed solutions to find the sum of different geometric sequences. Learn the key formulas and concepts behind the summation of sequences with common ratios greater and less than 1. Perfect for high school students in Grades 9-12.

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