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The image contains problems related to geometric sequences. Let's go through each question step-by-step.

### 1. Determine if the sequence is geometric. If it is, find the common ratio.
The sequences are:
1. \(1, -3, 9, -27, \ldots\)
2. \(5, 15, 45, 135, \ldots\)
3. \(\frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24}, \ldots\)
4. \(2, 5, 8, 11, \ldots\)

**Solution:**
- **(a)**
  - \( r = \frac{-3}{1} = -3, \; r = \frac{9}{-3} = -3 \) (Common ratio = -3)
  - Geometric sequence with common ratio \( r = -3 \).

- **(b)**
  - \( r = \frac{15}{5} = 3, \; r = \frac{45}{15} = 3 \) (Common ratio = 3)
  - Geometric sequence with common ratio \( r = 3 \).

- **(c)**
  - \( r = \frac{1/6}{1/3} = \frac{1}{2}, \; r = \frac{1/12}{1/6} = \frac{1}{2} \) (Common ratio = 1/2)
  - Geometric sequence with common ratio \( r = \frac{1}{2} \).

- **(d)**
  - Difference between terms: \( 5 - 2 = 3, \; 8 - 5 = 3, \; 11 - 8 = 3 \) (Common difference = 3)
  - Not a geometric sequence.

### 2. Given the explicit formula of a geometric sequence, find the first five terms.
Explicit formulas:
1. \( a_n = 5(-2)^{n-1} \)
2. \( a_n = \frac{2}{3} (4)^{n-1} \)

**Solution:**
- **(a)**  
  \( a_1 = 5(-2)^0 = 5, \; a_2 = 5(-2)^1 = -10, \; a_3 = 5(-2)^2 = 20, \; a_4 = 5(-2)^3 = -40, \; a_5 = 5(-2)^4 = 80 \)  
  First five terms: \( 5, -10, 20, -40, 80 \).

- **(b)**  
  \( a_1 = \frac{2}{3} (4)^0 = \frac{2}{3}, \; a_2 = \frac{2}{3} (4)^1 = \frac{8}{3}, \; a_3 = \frac{2}{3} (4)^2 = \frac{32}{3}, \; a_4 = \frac{2}{3} (4)^3 = \frac{128}{3}, \; a_5 = \frac{2}{3} (4)^4 = \frac{512}{3} \)  
  First five terms: \( \frac{2}{3}, \frac{8}{3}, \frac{32}{3}, \frac{128}{3}, \frac{512}{3} \).

### 3. Find the indicated term of the geometric sequence.
Given sequences:
1. \( a = 4, r = 3, n = 7 \)
2. \( a = 5, r = -2, n = 8 \)

**Solution:**
- **(a)** \( a_n = ar^{n-1} = 4 \times 3^{6} = 4 \times 729 = 2916 \)
- **(b)** \( a_n = ar^{n-1} = 5 \times (-2)^{7} = 5 \times (-128) = -640 \)

### 4. Find the sum, \( S_n \), for the first \( n \) terms of the given geometric sequence.
Given sequences:
1. \( a = 2, r = 3, n = 5 \)
2. \( a = 6, r = \frac{1}{2}, n = 6 \)

**Solution:**
- **(a)**  
  \( S_n = a \frac{r^n - 1}{r - 1} = 2 \frac{3^5 - 1}{2} = 2 \frac{243 - 1}{2} = 2 \times 121 = 242 \)

- **(b)**  
  \( S_n = a \frac{1 - r^n}{1 - r} = 6 \frac{1 - (\frac{1}{2})^6}{\frac{1}{2}} = 6 \frac{1 - \frac{1}{64}}{\frac{1}{2}} = 6 \times \frac{\frac{63}{64}}{\frac{1}{2}} = 6 \times \frac{63}{32} = \frac{378}{32} = 11.8125 \)

### 5. Find the indicated geometric sum.
Given:
1. \(\sum_{k=1}^{8} 2 \cdot (3)^{k-1}\)
2. \(\sum_{k=1}^{5} 5 \cdot \left(\frac{1}{2}\right)^{k-1}\)

**Solution:**
- **(a)**  
  \( S_8 = 2 \frac{3^8 - 1}{3 - 1} = 2 \frac{6561 - 1}{2} = 2 \times 3280 = 6560 \)

- **(b)**  
  \( S_5 = 5 \frac{1 - \left(\frac{1}{2}\right)^5}{1 - \frac{1}{2}} = 5 \frac{1 - \frac{1}{32}}{\frac{1}{2}} = 5 \times \frac{\frac{31}{32}}{\frac{1}{2}} = 5 \times \frac{31}{16} = \frac{155}{16} = 9.6875 \)

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### Relative Questions
1. How do you determine if a sequence is geometric?
2. What is the formula for the \(n\)-th term of a geometric sequence?
3. How can you find the sum of an infinite geometric series?
4. What is the significance of the common ratio in a geometric sequence?
5. How does the explicit formula of a geometric sequence differ from its recursive formula?

### Tip
When working with geometric sequences, always check the common ratio by dividing consecutive terms to confirm the sequence's geometric nature.

Determine if the sequence is geometric. If it is, find the common ratio. Given the explicit formula of a geometric sequence, find the first five terms. Find the indicated term of the geometric sequence. Find the sum for the first terms of the given geometric sequence. Find the indicated geometric sum.

Learn how to determine if a sequence is geometric, find the common ratio, and calculate the sum of a geometric series. This guide provides solutions for identifying geometric sequences, finding the nth term, and calculating the sum of the first n terms. Suitable for high school students (Grades 9-12).

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