To identify which sequences are geometric, we need to check if each sequence has a common ratio between consecutive terms. The common ratio \(r\) can be found by dividing any term in the sequence by the preceding term. If this ratio is constant for all consecutive terms in the sequence, then the sequence is geometric.
Let's examine each sequence provided:
1. **Sequence: –2.7, –9, –30, –100, ...**
\[ \frac{-9}{-2.7} = \frac{9}{2.7} \approx 3.33 \]
\[ \frac{-30}{-9} = \frac{30}{9} \approx 3.33 \]
\[ \frac{-100}{-30} = \frac{100}{30} \approx 3.33 \]
The ratio is approximately constant, so this sequence appears to be geometric.
2. **Sequence: –1, 2.5, –6.25, 15.625, ...**
\[ \frac{2.5}{-1} = -2.5 \]
\[ \frac{-6.25}{2.5} = -2.5 \]
\[ \frac{15.625}{-6.25} = -2.5 \]
The ratio is constant, so this sequence is geometric.
3. **Sequence: 9.1, 9.2, 9.3, 9.4, ...**
\[ \frac{9.2}{9.1} \approx 1.010989 \]
\[ \frac{9.3}{9.2} \approx 1.1087 \]
\[ \frac{9.4}{9.3} \approx 1.01075 \]
The ratios are not exactly constant, so this sequence is not geometric.
4. **Sequence: 8, 0.8, 0.08, 0.008, ...**
\[ \frac{0.8}{8} = 0.1 \]
\[ \frac{0.08}{0.8} = 0.1 \]
\[ \frac{0.008}{0.08} = 0.1 \]
The ratio is constant, so this sequence is geometric.
5. **Sequence: 4, –4, –12, –20, ...**
\[ \frac{-4}{4} = -1 \]
\[ \frac{-12}{-4} = 3 \]
\[ \frac{-20}{-12} = 1.67 \]
The ratios are not constant, so this sequence is not geometric.
The geometric sequences are:
- –2.7, –9, –30, –100, ...
- –1, 2.5, –6.25, 15.625, ...
- 8, 0.8, 0.08, 0.008, ...
Which sequences are geometric? Select three options.
–2.7, –9, –30, –100, ...
–1, 2.5, –6.25, 15.625, ...
9.1, 9.2, 9.3, 9.4, ...
8, 0.8, 0.08, 0.008, ...
4, –4, –12, –20,
1 answer
2 answers