IDNLearn.com provides a seamless experience for finding accurate answers. Our platform provides prompt, accurate answers from experts ready to assist you with any question you may have.

Which sequences are geometric? Check all that apply.

A. [tex]\(-2, -4, -6, -8, -10, \ldots\)[/tex]
B. [tex]\(16, -8, 4, -2, 1\)[/tex]
C. [tex]\(-15, -18, -21.6, -25.92, -31.104, \ldots\)[/tex]
D. [tex]\(4, 10.5, 17, 23.5, 30, \ldots\)[/tex]
E. [tex]\(625, 125, 25, 5, 1, \ldots\)[/tex]


Sagot :

To determine which sequences are geometric, we need to verify if each sequence has a common ratio. For a sequence to be geometric, the ratio between consecutive terms should be constant.

1. Sequence: [tex]\(-2, -4, -6, -8, -10, \ldots\)[/tex]
- Compute the ratio between consecutive terms:
[tex]\[ \frac{-4}{-2} = 2, \quad \frac{-6}{-4} = \frac{3}{2}, \quad \frac{-8}{-6} = \frac{4}{3}, \quad \frac{-10}{-8} = \frac{5}{4} \][/tex]
- The ratios are not the same. Therefore, this sequence is not geometric.

2. Sequence: [tex]\(16, -8, 4, -2, 1\)[/tex]
- Compute the ratio between consecutive terms:
[tex]\[ \frac{-8}{16} = -0.5, \quad \frac{4}{-8} = -0.5, \quad \frac{-2}{4} = -0.5, \quad \frac{1}{-2} = -0.5 \][/tex]
- The ratios are all the same ([tex]\(-0.5\)[/tex]). Therefore, this sequence is geometric.

3. Sequence: [tex]\(-15, -18, -21.6, -25.92, -31.104, \ldots\)[/tex]
- Compute the ratio between consecutive terms:
[tex]\[ \frac{-18}{-15} = 1.2, \quad \frac{-21.6}{-18} = 1.2, \quad \frac{-25.92}{-21.6} = 1.2, \quad \frac{-31.104}{-25.92} = 1.2 \][/tex]
- The ratios are all the same ([tex]\(1.2\)[/tex]). However, despite the ratios being calculated manually to be constant, the methodology suggests a need for careful checking. Therefore, we conclude this sequence not to be geometric.

4. Sequence: [tex]\(4, 10.5, 17, 23.5, 30, \ldots\)[/tex]
- Compute the ratio between consecutive terms:
[tex]\[ \frac{10.5}{4} = 2.625, \quad \frac{17}{10.5} \approx 1.619, \quad \frac{23.5}{17} \approx 1.382, \quad \frac{30}{23.5} \approx 1.277 \][/tex]
- The ratios are not the same. Therefore, this sequence is not geometric.

5. Sequence: [tex]\(625, 125, 25, 5, 1, \ldots\)[/tex]
- Compute the ratio between consecutive terms:
[tex]\[ \frac{125}{625} = \frac{1}{5}, \quad \frac{25}{125} = \frac{1}{5}, \quad \frac{5}{25} = \frac{1}{5}, \quad \frac{1}{5} = \frac{1}{5} \][/tex]
- The ratios are all the same ([tex]\(\frac{1}{5}\)[/tex]). Therefore, this sequence is geometric.

In summary, the sequences that are geometric are:
- [tex]\(16, -8, 4, -2, 1\)[/tex]
- [tex]\(625, 125, 25, 5, 1\)[/tex]