To determine which of the given sequences are geometric, we need to check if there is a constant ratio between consecutive terms in each sequence.
1. Sequence: [tex]\(-2.7, -9, -30, -100\)[/tex]
- Calculate the ratio between each pair of consecutive terms:
- [tex]\(\frac{-9}{-2.7} \approx 3.33\)[/tex]
- [tex]\(\frac{-30}{-9} \approx 3.33\)[/tex]
- [tex]\(\frac{-100}{-30} \approx 3.33\)[/tex]
- The ratios are consistent, so this sequence is not geometric.
2. Sequence: [tex]\(-1, 2.5, -6.25, 15.625\)[/tex]
- Calculate the ratio between each pair of consecutive terms:
- [tex]\(\frac{2.5}{-1} = -2.5\)[/tex]
- [tex]\(\frac{-6.25}{2.5} = -2.5\)[/tex]
- [tex]\(\frac{15.625}{-6.25} = -2.5\)[/tex]
- The ratios are consistent, so this sequence is geometric.
3. Sequence: [tex]\(9.1, 9.2, 9.3, 9.4\)[/tex]
- Calculate the ratio between each pair of consecutive terms:
- [tex]\(\frac{9.2}{9.1} \approx 1.01\)[/tex]
- [tex]\(\frac{9.3}{9.2} \approx 1.01\)[/tex]
- [tex]\(\frac{9.4}{9.3} \approx 1.01\)[/tex]
- The ratios are consistent, so this sequence is not geometric.
4. Sequence: [tex]\(8, 0.8, 0.08, 0.008\)[/tex]
- Calculate the ratio between each pair of consecutive terms:
- [tex]\(\frac{0.8}{8} = 0.1\)[/tex]
- [tex]\(\frac{0.08}{0.8} = 0.1\)[/tex]
- [tex]\(\frac{0.008}{0.08} = 0.1\)[/tex]
- The ratios are consistent, so this sequence is not geometric.
5. Sequence: [tex]\(4, -4, -12, -20\)[/tex]
- Calculate the ratio between each pair of consecutive terms:
- [tex]\(\frac{-4}{4} = -1\)[/tex]
- [tex]\(\frac{-12}{-4} = 3\)[/tex]
- [tex]\(\frac{-20}{-12} \approx 1.67\)[/tex]
- The ratios are not consistent, so this sequence is not geometric.
Based on the analysis, the sequence that is geometric is:
- [tex]\(-1, 2.5, -6.25, 15.625\)[/tex]
So, the correct answer with the geometric sequence is:
- [tex]\(-1, 2.5, -6.25, 15.625\)[/tex]
