https://file.aiquickdraw.com/mathbot/file/1726058246574z6s0hmwt.jpg

The image displays three geometric sequences, and the task is to find their common ratios and calculate the sum of the series for the given number of terms. Let's break down the sequences and solve them:

1. **Sequence 1**: \( 4, 12, 36, 108, \ldots \) (up to 7 terms)
   - The first term \( a = 4 \).
   - The common ratio \( r = \frac{12}{4} = 3 \).
   - Formula for the sum of a geometric series: 
     \[
     S_n = \frac{a(1 - r^n)}{1 - r}, \quad r \neq 1
     \]
     For 7 terms:
     \[
     S_7 = \frac{4(1 - 3^7)}{1 - 3} = \frac{4(1 - 2187)}{-2} = \frac{4(-2186)}{-2} = 4372
     \]
     So, the sum of the first 7 terms is **4372**.

2. **Sequence 2**: \( 2, 6, 18, 54, 162, \ldots \) (up to 6 terms)
   - The first term \( a = 2 \).
   - The common ratio \( r = \frac{6}{2} = 3 \).
   - Using the same formula for 6 terms:
     \[
     S_6 = \frac{2(1 - 3^6)}{1 - 3} = \frac{2(1 - 729)}{-2} = \frac{2(-728)}{-2} = 728
     \]
     So, the sum of the first 6 terms is **728**.

3. **Sequence 3**: \( 7, 14, 28, 56, \ldots \) (up to 8 terms)
   - The first term \( a = 7 \).
   - The common ratio \( r = \frac{14}{7} = 2 \).
   - For 8 terms:
     \[
     S_8 = \frac{7(1 - 2^8)}{1 - 2} = \frac{7(1 - 256)}{-1} = \frac{7(-255)}{-1} = 1785
     \]
     So, the sum of the first 8 terms is **1785**.

Let me know if you need any more details or explanations!

### Related Questions:
1. How do you derive the formula for the sum of a geometric series?
2. What happens when the common ratio is between 0 and 1?
3. How would you find the sum if the series were infinite?
4. Can a geometric series have a negative common ratio? How would it affect the sum?
5. How do you calculate the sum of an arithmetic series?

### Tip:
In geometric series, if the common ratio \( |r| < 1 \), the series can converge to a finite value, even if there are infinitely many terms!

Determine the common ratio and find the sum of the following geometric sequences: 1. 4, 12, 36, 108,... up to 7 terms. 2. 2, 6, 18, 54, 162,... up to 6 terms. 3. 7, 14, 28, 56,... up to 8 terms.

Learn how to find the common ratio and calculate the sum of geometric sequences for 7, 6, and 8 terms. Detailed breakdowns of each sequence, with applications of the geometric series formula. Suitable for high school students in grades 9-12.

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