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.