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Sagot :
Let's solve this step-by-step for each given geometric sequence. We'll determine both the growth factor and the next term in the sequence.
### Sequence a: [tex]\(1.5, 3, 6\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{3}{1.5} = 2.0 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = 6 \times 2.0 = 12.0 \][/tex]
### Sequence b: [tex]\(40, 120, 360\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{120}{40} = 3.0 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = 360 \times 3.0 = 1080.0 \][/tex]
### Sequence c: [tex]\(200, 20, 2\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{20}{200} = 0.1 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = 2 \times 0.1 = 0.2 \][/tex]
### Sequence d: [tex]\(-\frac{1}{7}, \frac{9}{7}, \frac{27}{7}\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{\frac{9}{7}}{-\frac{1}{7}} = -9.0 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = \frac{27}{7} \times -9.0 = -\frac{243}{7} = -34.71428571428572 \][/tex]
### Sequence e: [tex]\(24, 12, 6\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{12}{24} = 0.5 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = 6 \times 0.5 = 3.0 \][/tex]
### Summary
- The sequences and their next terms are:
1. [tex]\(1.5, 3, 6, 12.0\)[/tex] with a growth factor of [tex]\(2.0\)[/tex]
2. [tex]\(40, 120, 360, 1080.0\)[/tex] with a growth factor of [tex]\(3.0\)[/tex]
3. [tex]\(200, 20, 2, 0.2\)[/tex] with a growth factor of [tex]\(0.1\)[/tex]
4. [tex]\(-\frac{1}{7}, \frac{9}{7}, \frac{27}{7}, -34.71428571428572\)[/tex] with a growth factor of [tex]\(-9.0\)[/tex]
5. [tex]\(24, 12, 6, 3.0\)[/tex] with a growth factor of [tex]\(0.5\)[/tex]
I hope this detailed solution helps you understand how to complete each geometric sequence and find the respective growth factors!
### Sequence a: [tex]\(1.5, 3, 6\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{3}{1.5} = 2.0 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = 6 \times 2.0 = 12.0 \][/tex]
### Sequence b: [tex]\(40, 120, 360\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{120}{40} = 3.0 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = 360 \times 3.0 = 1080.0 \][/tex]
### Sequence c: [tex]\(200, 20, 2\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{20}{200} = 0.1 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = 2 \times 0.1 = 0.2 \][/tex]
### Sequence d: [tex]\(-\frac{1}{7}, \frac{9}{7}, \frac{27}{7}\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{\frac{9}{7}}{-\frac{1}{7}} = -9.0 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = \frac{27}{7} \times -9.0 = -\frac{243}{7} = -34.71428571428572 \][/tex]
### Sequence e: [tex]\(24, 12, 6\)[/tex]
1. Growth Factor:
- Find the ratio between successive terms:
[tex]\[ \text{Growth Factor} = \frac{12}{24} = 0.5 \][/tex]
2. Next Term:
- Multiply the last term by the growth factor:
[tex]\[ \text{Next Term} = 6 \times 0.5 = 3.0 \][/tex]
### Summary
- The sequences and their next terms are:
1. [tex]\(1.5, 3, 6, 12.0\)[/tex] with a growth factor of [tex]\(2.0\)[/tex]
2. [tex]\(40, 120, 360, 1080.0\)[/tex] with a growth factor of [tex]\(3.0\)[/tex]
3. [tex]\(200, 20, 2, 0.2\)[/tex] with a growth factor of [tex]\(0.1\)[/tex]
4. [tex]\(-\frac{1}{7}, \frac{9}{7}, \frac{27}{7}, -34.71428571428572\)[/tex] with a growth factor of [tex]\(-9.0\)[/tex]
5. [tex]\(24, 12, 6, 3.0\)[/tex] with a growth factor of [tex]\(0.5\)[/tex]
I hope this detailed solution helps you understand how to complete each geometric sequence and find the respective growth factors!
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