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To determine which sequence shows a pattern where each term is 1.5 times the previous term, we will analyze the given sequences step-by-step.
### Sequence 1: [tex]\(-4, 6, -9, 13.5, \ldots\)[/tex]
1. Ratio between second term and first term: [tex]\( \frac{6}{-4} = -1.5 \)[/tex]
2. Ratio between third term and second term: [tex]\( \frac{-9}{6} = -1.5 \)[/tex]
3. Ratio between fourth term and third term: [tex]\( \frac{13.5}{-9} = -1.5 \)[/tex]
The ratios are all [tex]\(-1.5\)[/tex]. Therefore, this sequence does not show a pattern where each term is [tex]\( 1.5 \)[/tex] times the previous term.
### Sequence 2: [tex]\(10, 15, 25, 40, \ldots\)[/tex]
1. Ratio between second term and first term: [tex]\( \frac{15}{10} = 1.5 \)[/tex]
2. Ratio between third term and second term: [tex]\( \frac{25}{15} \approx 1.67 \)[/tex]
3. Ratio between fourth term and third term: [tex]\( \frac{40}{25} = 1.6 \)[/tex]
The ratios are [tex]\(1.5\)[/tex], [tex]\(1.67\)[/tex], and [tex]\(1.6\)[/tex], which are not consistent. Hence, this sequence does not show a pattern where each term is [tex]\(1.5\)[/tex] times the previous term.
### Sequence 3: [tex]\(98, 99.5, 101, 102.5, \ldots\)[/tex]
1. Ratio between second term and first term: [tex]\( \frac{99.5}{98} \approx 1.015 \)[/tex]
2. Ratio between third term and second term: [tex]\( \frac{101}{99.5} \approx 1.015 \)[/tex]
3. Ratio between fourth term and third term: [tex]\( \frac{102.5}{101} \approx 1.015 \)[/tex]
The ratios are approximately [tex]\( 1.015 \)[/tex]. Therefore, this sequence does not show a pattern where each term is [tex]\( 1.5 \)[/tex] times the previous term.
### Sequence 4: [tex]\(-200, -300, -450, -675, \ldots\)[/tex]
1. Ratio between second term and first term: [tex]\( \frac{-300}{-200} = 1.5 \)[/tex]
2. Ratio between third term and second term: [tex]\( \frac{-450}{-300} = 1.5 \)[/tex]
3. Ratio between fourth term and third term: [tex]\( \frac{-675}{-450} = 1.5 \)[/tex]
All ratios are consistently [tex]\(1.5\)[/tex]. Thus, this sequence shows a pattern where each term is [tex]\(1.5\)[/tex] times the previous term.
### Conclusion
The sequence that shows a pattern where each term is [tex]\( 1.5 \)[/tex] times the previous term is:
[tex]\[ -200, -300, -450, -675, \ldots \][/tex]
Hence, the correct sequence is Sequence 4.
### Sequence 1: [tex]\(-4, 6, -9, 13.5, \ldots\)[/tex]
1. Ratio between second term and first term: [tex]\( \frac{6}{-4} = -1.5 \)[/tex]
2. Ratio between third term and second term: [tex]\( \frac{-9}{6} = -1.5 \)[/tex]
3. Ratio between fourth term and third term: [tex]\( \frac{13.5}{-9} = -1.5 \)[/tex]
The ratios are all [tex]\(-1.5\)[/tex]. Therefore, this sequence does not show a pattern where each term is [tex]\( 1.5 \)[/tex] times the previous term.
### Sequence 2: [tex]\(10, 15, 25, 40, \ldots\)[/tex]
1. Ratio between second term and first term: [tex]\( \frac{15}{10} = 1.5 \)[/tex]
2. Ratio between third term and second term: [tex]\( \frac{25}{15} \approx 1.67 \)[/tex]
3. Ratio between fourth term and third term: [tex]\( \frac{40}{25} = 1.6 \)[/tex]
The ratios are [tex]\(1.5\)[/tex], [tex]\(1.67\)[/tex], and [tex]\(1.6\)[/tex], which are not consistent. Hence, this sequence does not show a pattern where each term is [tex]\(1.5\)[/tex] times the previous term.
### Sequence 3: [tex]\(98, 99.5, 101, 102.5, \ldots\)[/tex]
1. Ratio between second term and first term: [tex]\( \frac{99.5}{98} \approx 1.015 \)[/tex]
2. Ratio between third term and second term: [tex]\( \frac{101}{99.5} \approx 1.015 \)[/tex]
3. Ratio between fourth term and third term: [tex]\( \frac{102.5}{101} \approx 1.015 \)[/tex]
The ratios are approximately [tex]\( 1.015 \)[/tex]. Therefore, this sequence does not show a pattern where each term is [tex]\( 1.5 \)[/tex] times the previous term.
### Sequence 4: [tex]\(-200, -300, -450, -675, \ldots\)[/tex]
1. Ratio between second term and first term: [tex]\( \frac{-300}{-200} = 1.5 \)[/tex]
2. Ratio between third term and second term: [tex]\( \frac{-450}{-300} = 1.5 \)[/tex]
3. Ratio between fourth term and third term: [tex]\( \frac{-675}{-450} = 1.5 \)[/tex]
All ratios are consistently [tex]\(1.5\)[/tex]. Thus, this sequence shows a pattern where each term is [tex]\(1.5\)[/tex] times the previous term.
### Conclusion
The sequence that shows a pattern where each term is [tex]\( 1.5 \)[/tex] times the previous term is:
[tex]\[ -200, -300, -450, -675, \ldots \][/tex]
Hence, the correct sequence is Sequence 4.
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