To determine which sequences are geometric, we need to check if each sequence has a common ratio between consecutive terms.
1. For the sequence [tex]\(-2.7, -9, -30, -100, \ldots\)[/tex]:
- Calculate the ratios between consecutive terms:
[tex]\[
\frac{-9}{-2.7} \approx 3.33, \quad \frac{-30}{-9} \approx 3.33, \quad \frac{-100}{-30} \approx 3.33
\][/tex]
- The ratios are not exactly equal, thus this sequence is not geometric.
2. For the sequence [tex]\(-1, 2.5, -6.25, 15.625, \ldots\)[/tex]:
- Calculate the ratios between consecutive terms:
[tex]\[
\frac{2.5}{-1} = -2.5, \quad \frac{-6.25}{2.5} = -2.5, \quad \frac{15.625}{-6.25} = -2.5
\][/tex]
- Since all the ratios are equal, this sequence is geometric.
3. For the sequence [tex]\(9.1, 9.2, 9.3, 9.4, \ldots\)[/tex]:
- Calculate the ratios between consecutive terms:
[tex]\[
\frac{9.2}{9.1} \approx 1.01, \quad \frac{9.3}{9.2} \approx 1.01, \quad \frac{9.4}{9.3} \approx 1.01
\][/tex]
- The ratios are close but not consistent. This sequence is not geometric.
4. For the sequence [tex]\(8, 0.8, 0.08, 0.008, \ldots\)[/tex]:
- Calculate the ratios between consecutive terms:
[tex]\[
\frac{0.8}{8} = 0.1, \quad \frac{0.08}{0.8} = 0.1, \quad \frac{0.008}{0.08} = 0.1
\][/tex]
- Since all the ratios are equal, this sequence is geometric.
5. For the sequence [tex]\(4, -4, -12, -20, \ldots\)[/tex]:
- Calculate the ratios between consecutive terms:
[tex]\[
\frac{-4}{4} = -1, \quad \frac{-12}{-4} = 3, \quad \frac{-20}{-12} \approx 1.67
\][/tex]
- The ratios are not consistent, thus this sequence is not geometric.
Based on the calculations, the sequences that are geometric are:
[tex]\[ -1, 2.5, -6.25, 15.625, \ldots \][/tex]
[tex]\[ 8, 0.8, 0.08, 0.008, \ldots \][/tex]
Thus, the correct sequences are:
[tex]\[ \boxed{2} \][/tex]
