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Instructions: For each of the sequences below, identify whether there is a common ratio. If there is, identify what it is. If there is not a common ratio, type "none."

[tex]\[
\begin{array}{l}
1. \ 5, \ 10, \ 15, \ 20, \ 25 \\
2. \ 1, \ 2, \ 4, \ 8, \ 16 \\
3. \ 3, \ -9, \ 27, \ -81 \\
4. \ 10, \ 7, \ 4, \ 1, \ -2 \\
5. \ 1, \ 10, \ 100, \ 1000 \\
6. \ 10, \ 5, \ 2.5, \ 1.25 \\
\end{array}
\][/tex]

1. none
2. none
3. [tex]\(-3\)[/tex]
4. none
5. [tex]\(10\)[/tex]
6. [tex]\(\frac{1}{2}\)[/tex]



Answer :

GinnyAnswer
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]
```

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