Comprehensive worksheet on geometric sequences covering identification, common ratio calculation, sequence completion, and examples.

Problem Analysis and Solution

The worksheet focuses on geometric sequences. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

#### Section A: Circle all the geometric sequences below

We need to identify which of the given sequences are geometric. A sequence is geometric if the ratio between consecutive terms is constant.

1. 1, 1, 2, 3, 5, 8, ...
- Ratios: \( \frac{1}{1} = 1 \), \( \frac{2}{1} = 2 \), \( \frac{3}{2} = 1.5 \), \( \frac{5}{3} \approx 1.67 \)
- Not geometric (ratios are not constant).

2. 6000, 3000, 1500, ...
- Ratios: \( \frac{3000}{6000} = \frac{1}{2} \), \( \frac{1500}{3000} = \frac{1}{2} \)
- Geometric (common ratio = \( \frac{1}{2} \)).

3. 1, 3, 6, 10, 15, ...
- Ratios: \( \frac{3}{1} = 3 \), \( \frac{6}{3} = 2 \), \( \frac{10}{6} \approx 1.67 \)
- Not geometric (ratios are not constant).

4. 1, \( \frac{1}{3} \), \( \frac{1}{9} \), \( \frac{1}{27} \), ...
- Ratios: \( \frac{\frac{1}{3}}{1} = \frac{1}{3} \), \( \frac{\frac{1}{9}}{\frac{1}{3}} = \frac{1}{3} \)
- Geometric (common ratio = \( \frac{1}{3} \)).

5. -8, -16, -32, -64, ...
- Ratios: \( \frac{-16}{-8} = 2 \), \( \frac{-32}{-16} = 2 \)
- Geometric (common ratio = 2).

6. \( x, x+1, x+2, x+3, \ldots \)
- Ratios: \( \frac{x+1}{x} \neq \frac{x+2}{x+1} \)
- Not geometric (ratios are not constant).

7. 10, 100, 1000, 10000, ...
- Ratios: \( \frac{100}{10} = 10 \), \( \frac{1000}{100} = 10 \)
- Geometric (common ratio = 10).

8. -1, 1, -1, 1, -1, ...
- Ratios: \( \frac{1}{-1} = -1 \), \( \frac{-1}{1} = -1 \)
- Geometric (common ratio = -1).

9. 5, 10, 15, 20, ...
- Ratios: \( \frac{10}{5} = 2 \), \( \frac{15}{10} = 1.5 \)
- Not geometric (ratios are not constant).

10. 0.1, 0.2, 0.3, 0.4, ...
- Ratios: \( \frac{0.2}{0.1} = 2 \), \( \frac{0.3}{0.2} = 1.5 \)
- Not geometric (ratios are not constant).

11. 4, 6, 9, 13.5, ...
- Ratios: \( \frac{6}{4} = 1.5 \), \( \frac{9}{6} = 1.5 \)
- Geometric (common ratio = 1.5).

12. \( a, 2a, 4a, 8a, \ldots \)
- Ratios: \( \frac{2a}{a} = 2 \), \( \frac{4a}{2a} = 2 \)
- Geometric (common ratio = 2).

Geometric sequences:
- 2, 4, 5, 7, 8, 11, 12

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Section B: Find the common ratio of the geometric sequences

1. 5, 20, 80, 320, ...
- Common ratio: \( \frac{20}{5} = 4 \)

2. 1, -5, 25, -125, 625, ...
- Common ratio: \( \frac{-5}{1} = -5 \)

3. 3, 4.5, 6.75, 10.125, ...
- Common ratio: \( \frac{4.5}{3} = 1.5 \)

4. 3.2, 6.4, 12.8, 25.6, ...
- Common ratio: \( \frac{6.4}{3.2} = 2 \)

5. 6000, 600, 60, 6, ...
- Common ratio: \( \frac{600}{6000} = \frac{1}{10} \)

6. 1, ?, 9, ?, 81, ...
- Common ratio: \( \sqrt{\frac{9}{1}} = 3 \) (since it's a geometric sequence, the ratio must be consistent).

7. 1, \( \frac{1}{3} \), \( \frac{1}{9} \), \( \frac{1}{27} \), ...
- Common ratio: \( \frac{\frac{1}{3}}{1} = \frac{1}{3} \)

8. 10, 2, 0.4, 0.125, ...
- Common ratio: \( \frac{2}{10} = \frac{1}{5} \)

9. \( x, x^2, x^3, x^4, \ldots \)
- Common ratio: \( \frac{x^2}{x} = x \)

10. -7, -14, -28, -56, -112, ...
- Common ratio: \( \frac{-14}{-7} = 2 \)

Answers:
1. 4
2. -5
3. 1.5
4. 2
5. \( \frac{1}{10} \)
6. 3
7. \( \frac{1}{3} \)
8. \( \frac{1}{5} \)
9. \( x \)
10. 2

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Section C: Fill the gaps in these geometric sequences

1. 2, \_\_\_, 200, \_\_\_, 20000, ...
- Common ratio: \( \sqrt{\frac{200}{2}} = 10 \)
- Missing terms: \( 2 \times 10 = 20 \), \( 200 \times 10 = 2000 \)
- Sequence: 2, 20, 200, 2000, 20000

2. \_\_\_, 15, 75, \_\_\_, ...
- Common ratio: \( \frac{75}{15} = 5 \)
- Missing terms: \( \frac{15}{5} = 3 \), \( 75 \times 5 = 375 \)
- Sequence: 3, 15, 75, 375

3. 1, 4, \_\_\_, \_\_\_, ...
- Common ratio: \( \frac{4}{1} = 4 \)
- Missing terms: \( 4 \times 4 = 16 \), \( 16 \times 4 = 64 \)
- Sequence: 1, 4, 16, 64

4. 7, \_\_\_, \_\_\_, 189, ...
- Common ratio: \( \sqrt[3]{\frac{189}{7}} = 3 \)
- Missing terms: \( 7 \times 3 = 21 \), \( 21 \times 3 = 63 \)
- Sequence: 7, 21, 63, 189

5. 200, \_\_\_, 50, \_\_\_, ...
- Common ratio: \( \sqrt{\frac{50}{200}} = \frac{1}{2} \)
- Missing terms: \( 200 \times \frac{1}{2} = 100 \), \( 50 \times \frac{1}{2} = 25 \)
- Sequence: 200, 100, 50, 25

6. \_\_\_, 12, -36, \_\_\_, ...
- Common ratio: \( \frac{-36}{12} = -3 \)
- Missing terms: \( \frac{12}{-3} = -4 \), \( -36 \times -3 = 108 \)
- Sequence: -4, 12, -36, 108

7. 8, \_\_\_, 8, \_\_\_, ...
- Common ratio: \( \sqrt{\frac{8}{8}} = 1 \)
- Missing terms: \( 8 \times 1 = 8 \), \( 8 \times 1 = 8 \)
- Sequence: 8, 8, 8, 8

8. \( \frac{1}{3} \), \_\_\_, \( \frac{1}{12} \), \_\_\_, ...
- Common ratio: \( \sqrt{\frac{\frac{1}{12}}{\frac{1}{3}}} = \frac{1}{2} \)
- Missing terms: \( \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \), \( \frac{1}{12} \times \frac{1}{2} = \frac{1}{24} \)
- Sequence: \( \frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24} \)

9. 4096, 512, \_\_\_, 8, \_\_\_, ...
- Common ratio: \( \frac{512}{4096} = \frac{1}{8} \)
- Missing terms: \( 512 \times \frac{1}{8} = 64 \), \( 8 \times \frac{1}{8} = 1 \)
- Sequence: 4096, 512, 64, 8, 1

10. -20, -100, \_\_\_, \_\_\_, ...
- Common ratio: \( \frac{-100}{-20} = 5 \)
- Missing terms: \( -100 \times 5 = -500 \), \( -500 \times 5 = -2500 \)
- Sequence: -20, -100, -500, -2500

Answers:
1. 20, 2000
2. 3, 375
3. 16, 64
4. 21, 63
5. 100, 25
6. -4, 108
7. 8, 8
8. \( \frac{1}{6}, \frac{1}{24} \)
9. 64, 1
10. -500, -2500

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Section D: Show me...

1. A sequence with a common ratio of 6
- Example: \( 1, 6, 36, 216, \ldots \)

2. A decreasing geometric sequence
- Example: \( 100, 50, 25, 12.5, \ldots \) (common ratio = \( \frac{1}{2} \))

3. A sequence with a common ratio of -2
- Example: \( 1, -2, 4, -8, \ldots \)

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Final Answer

\[
\boxed{
\text{Section A: 2, 4, 5, 7, 8, 11, 12}
}
\]
\[
\boxed{
\text{Section B: 4, -5, 1.5, 2, } \frac{1}{10}, 3, \frac{1}{3}, \frac{1}{5}, x, 2
}
\]
\[
\boxed{
\text{Section C: 20, 2000; 3, 375; 16, 64; 21, 63; 100, 25; -4, 108; 8, 8; } \frac{1}{6}, \frac{1}{24}; 64, 1; -500, -2500
}
\]
\[
\boxed{
\text{Section D: 1, 6, 36, 216, ...; 100, 50, 25, 12.5, ...; 1, -2, 4, -8, ...}
}
\]