Let's analyze each sequence to determine if there is a common ratio and if so, identify it.
1. Sequence: [tex]\(5, 10, 15, 20, 25\)[/tex]
- To identify a common ratio, we divide each term by the preceding term.
- [tex]\(\frac{10}{5} = 2,\)[/tex]
- [tex]\(\frac{15}{10} = 1.5,\)[/tex]
- [tex]\(\frac{20}{15} \approx 1.333,\)[/tex]
- [tex]\(\frac{25}{20} = 1.25.\)[/tex]
Since the ratios are not constant, this sequence does not have a common ratio. Thus, the answer is none.
2. Sequence: [tex]\(1, 2, 4, 8, 16\)[/tex]
- [tex]\(\frac{2}{1} = 2,\)[/tex]
- [tex]\(\frac{4}{2} = 2,\)[/tex]
- [tex]\(\frac{8}{4} = 2,\)[/tex]
- [tex]\(\frac{16}{8} = 2.\)[/tex]
Since the ratio between consecutive terms is constant (2), the common ratio is 2.
3. Sequence: [tex]\(3, -9, 27, -81\)[/tex]
- [tex]\(\frac{-9}{3} = -3,\)[/tex]
- [tex]\(\frac{27}{-9} = -3,\)[/tex]
- [tex]\(\frac{-81}{27} = -3.\)[/tex]
Since the ratio between consecutive terms is constant (-3), the common ratio is -3.
4. Sequence: [tex]\(10, 7, 4, 1, -2\)[/tex]
- [tex]\(\frac{7}{10} = 0.7,\)[/tex]
- [tex]\(\frac{4}{7} \approx 0.571,\)[/tex]
- [tex]\(\frac{1}{4} = 0.25,\)[/tex]
- [tex]\(\frac{-2}{1} = -2.\)[/tex]
Since the ratios are not constant, this sequence does not have a common ratio. Thus, the answer is none.
5. Sequence: [tex]\(1, 10, 100, 1000\)[/tex]
- [tex]\(\frac{10}{1} = 10,\)[/tex]
- [tex]\(\frac{100}{10} = 10,\)[/tex]
- [tex]\(\frac{1000}{100} = 10.\)[/tex]
Since the ratio between consecutive terms is constant (10), the common ratio is 10.
6. Sequence: [tex]\(10, 5, 2.5, 1.25\)[/tex]
- [tex]\(\frac{5}{10} = 0.5,\)[/tex]
- [tex]\(\frac{2.5}{5} = 0.5,\)[/tex]
- [tex]\(\frac{1.25}{2.5} = 0.5.\)[/tex]
Since the ratio between consecutive terms is constant (0.5), the common ratio is 0.5.
Let's match the results with the provided sequences:
[tex]\[
\begin{array}{l}
5,10,15,20,25 \quad \text{none} \\
1,2,4,8,16 \quad \text{2} \\
3,-9,27,-81 \quad \text{-3} \\
10,7,4,1,-2 \quad \text{none} \\
1,10,100,1000 \quad \text{10} \\
10,5,2.5,1.25 \quad \text{0.5}
\end{array}
\][/tex]
Thus, the common ratios identified for the sequences are:
- [tex]\(5, 10, 15, 20, 25: \quad \text{none}\)[/tex]
- [tex]\(1, 2, 4, 8, 16: \quad \text{2}\)[/tex]
- [tex]\(3, -9, 27, -81: \quad \text{-3}\)[/tex]
- [tex]\(10, 7, 4, 1, -2: \quad \text{none}\)[/tex]
- [tex]\(1, 10, 100, 1000: \quad \text{10}\)[/tex]
- [tex]\(10, 5, 2.5, 1.25: \quad \text{0.5}\)[/tex]
The final answer for the common ratios is:
```
[None, 2.0, -3.0, None, 10.0, 0.5]
```
