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Answer :
To determine which sequences are geometric, let's start by understanding what a geometric sequence is. A geometric sequence is a sequence in which each term after the first is found by multiplying the previous one by a constant called the common ratio.
We'll go through each sequence provided and check if it maintains the same common ratio throughout.
1. Sequence: [tex]\(20, 70, 245, 857.5, \ldots\)[/tex]
- To find if it's geometric, we check if the ratio between successive terms is constant.
- Calculate the ratio: [tex]\(\frac{70}{20} = 3.5\)[/tex], [tex]\(\frac{245}{70} = 3.5\)[/tex], [tex]\(\frac{857.5}{245} = 3.5\)[/tex]
- The ratio is consistent; thus, the sequence is geometric.
2. Sequence: [tex]\(13, 16.5, 20, 23.5, \ldots\)[/tex]
- Calculate the ratios between the terms: [tex]\(\frac{16.5}{13} \approx 1.269\)[/tex], [tex]\(\frac{20}{16.5} \approx 1.212\)[/tex]
- The ratios are not equal; therefore, this sequence is not geometric.
3. Sequence: [tex]\(160, 40, 10, 2.5, \ldots\)[/tex]
- Calculate the ratios: [tex]\(\frac{40}{160} = \frac{1}{4}\)[/tex], [tex]\(\frac{10}{40} = \frac{1}{4}\)[/tex], [tex]\(\frac{2.5}{10} = \frac{1}{4}\)[/tex]
- The ratio is consistent; hence, the sequence is geometric.
4. Sequence: [tex]\(5, 5.5, 6.05, 6.655, \ldots\)[/tex]
- Calculate the ratios: [tex]\(\frac{5.5}{5} = 1.1\)[/tex], [tex]\(\frac{6.05}{5.5} \approx 1.1\)[/tex]
- The ratio is consistent continuing as 1.1; so this sequence is geometric.
5. Sequence: [tex]\(16, 17.1, 18.2, 19.3, \ldots\)[/tex]
- Calculate the ratios: [tex]\(\frac{17.1}{16} \approx 1.069\)[/tex], [tex]\(\frac{18.2}{17.1} \approx 1.064\)[/tex]
- The ratios are not equal; therefore, this sequence is not geometric.
6. Sequence: [tex]\(7.5, 5.625, 4.21875\)[/tex]
- Calculate the ratios: [tex]\(\frac{5.625}{7.5} = 0.75\)[/tex], [tex]\(\frac{4.21875}{5.625} = 0.75\)[/tex]
- The ratio is consistent; thus, the sequence is geometric.
Based on these observations, the sequences that are geometric are:
1. [tex]\(20, 70, 245, 857.5, \ldots\)[/tex]
3. [tex]\(160, 40, 10, 2.5, \ldots\)[/tex]
4. [tex]\(5, 5.5, 6.05, 6.655, \ldots\)[/tex]
6. [tex]\(7.5, 5.625, 4.21875\)[/tex]
These correspond to the positions [1, 3, 4, 6] in the list.
We'll go through each sequence provided and check if it maintains the same common ratio throughout.
1. Sequence: [tex]\(20, 70, 245, 857.5, \ldots\)[/tex]
- To find if it's geometric, we check if the ratio between successive terms is constant.
- Calculate the ratio: [tex]\(\frac{70}{20} = 3.5\)[/tex], [tex]\(\frac{245}{70} = 3.5\)[/tex], [tex]\(\frac{857.5}{245} = 3.5\)[/tex]
- The ratio is consistent; thus, the sequence is geometric.
2. Sequence: [tex]\(13, 16.5, 20, 23.5, \ldots\)[/tex]
- Calculate the ratios between the terms: [tex]\(\frac{16.5}{13} \approx 1.269\)[/tex], [tex]\(\frac{20}{16.5} \approx 1.212\)[/tex]
- The ratios are not equal; therefore, this sequence is not geometric.
3. Sequence: [tex]\(160, 40, 10, 2.5, \ldots\)[/tex]
- Calculate the ratios: [tex]\(\frac{40}{160} = \frac{1}{4}\)[/tex], [tex]\(\frac{10}{40} = \frac{1}{4}\)[/tex], [tex]\(\frac{2.5}{10} = \frac{1}{4}\)[/tex]
- The ratio is consistent; hence, the sequence is geometric.
4. Sequence: [tex]\(5, 5.5, 6.05, 6.655, \ldots\)[/tex]
- Calculate the ratios: [tex]\(\frac{5.5}{5} = 1.1\)[/tex], [tex]\(\frac{6.05}{5.5} \approx 1.1\)[/tex]
- The ratio is consistent continuing as 1.1; so this sequence is geometric.
5. Sequence: [tex]\(16, 17.1, 18.2, 19.3, \ldots\)[/tex]
- Calculate the ratios: [tex]\(\frac{17.1}{16} \approx 1.069\)[/tex], [tex]\(\frac{18.2}{17.1} \approx 1.064\)[/tex]
- The ratios are not equal; therefore, this sequence is not geometric.
6. Sequence: [tex]\(7.5, 5.625, 4.21875\)[/tex]
- Calculate the ratios: [tex]\(\frac{5.625}{7.5} = 0.75\)[/tex], [tex]\(\frac{4.21875}{5.625} = 0.75\)[/tex]
- The ratio is consistent; thus, the sequence is geometric.
Based on these observations, the sequences that are geometric are:
1. [tex]\(20, 70, 245, 857.5, \ldots\)[/tex]
3. [tex]\(160, 40, 10, 2.5, \ldots\)[/tex]
4. [tex]\(5, 5.5, 6.05, 6.655, \ldots\)[/tex]
6. [tex]\(7.5, 5.625, 4.21875\)[/tex]
These correspond to the positions [1, 3, 4, 6] in the list.
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