Answer :
Certainly! Let’s classify the given sequences as Arithmetic, Geometric, or Neither, based on their defining properties.
### Step-by-Step Solution:
#### Sequence 1: [tex]\(-1, 3, 7, 11, 15, \ldots\)[/tex]
1. Identify if it is an Arithmetic Sequence:
- Check the differences between consecutive terms.
- [tex]\(3 - (-1) = 4\)[/tex]
- [tex]\(7 - 3 = 4\)[/tex]
- [tex]\(11 - 7 = 4\)[/tex]
- [tex]\(15 - 11 = 4\)[/tex]
- The difference between consecutive terms is constant (4).
2. Since the difference is constant, this sequence is Arithmetic.
#### Sequence 2: [tex]\(36, 12, 4, \frac{4}{3}, \frac{4}{9}, \ldots\)[/tex]
1. Identify if it is a Geometric Sequence:
- Check the ratios between consecutive terms.
- [tex]\(\frac{12}{36} = \frac{1}{3}\)[/tex]
- [tex]\(\frac{4}{12} = \frac{1}{3}\)[/tex]
- [tex]\(\frac{4/3}{4} = \frac{1}{3}\)[/tex]
- [tex]\(\frac{4/9}{4/3} = \frac{1}{3}\)[/tex]
- The ratio between consecutive terms is constant [tex]\(\left(\frac{1}{3}\right)\)[/tex].
2. Since the ratio is constant, this sequence is Geometric.
#### Sequence 3: [tex]\(-4, -2, 1, 4, 8, \ldots\)[/tex]
1. Check if it is an Arithmetic Sequence:
- Check the differences between consecutive terms.
- [tex]\(-2 - (-4) = 2\)[/tex]
- [tex]\(1 - (-2) = 3\)[/tex]
- [tex]\(4 - 1 = 3\)[/tex]
- [tex]\(8 - 4 = 4\)[/tex]
- The differences between consecutive terms are not constant.
2. Check if it is a Geometric Sequence:
- Check the ratios between consecutive terms.
- [tex]\(\frac{-2}{-4} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{1}{-2} = -\frac{1}{2}\)[/tex]
- [tex]\(\frac{4}{1} = 4\)[/tex]
- [tex]\(\frac{8}{4} = 2\)[/tex]
- The ratios between consecutive terms are not constant.
3. Since neither the differences nor the ratios are constant, this sequence is Neither.
### Classification:
- Arithmetic: [tex]\(-1, 3, 7, 11, 15, \ldots\)[/tex]
- Geometric: [tex]\(36, 12, 4, \frac{4}{3}, \frac{4}{9}, \ldots\)[/tex]
- Neither: [tex]\(-4, -2, 1, 4, 8, \ldots\)[/tex]
### Step-by-Step Solution:
#### Sequence 1: [tex]\(-1, 3, 7, 11, 15, \ldots\)[/tex]
1. Identify if it is an Arithmetic Sequence:
- Check the differences between consecutive terms.
- [tex]\(3 - (-1) = 4\)[/tex]
- [tex]\(7 - 3 = 4\)[/tex]
- [tex]\(11 - 7 = 4\)[/tex]
- [tex]\(15 - 11 = 4\)[/tex]
- The difference between consecutive terms is constant (4).
2. Since the difference is constant, this sequence is Arithmetic.
#### Sequence 2: [tex]\(36, 12, 4, \frac{4}{3}, \frac{4}{9}, \ldots\)[/tex]
1. Identify if it is a Geometric Sequence:
- Check the ratios between consecutive terms.
- [tex]\(\frac{12}{36} = \frac{1}{3}\)[/tex]
- [tex]\(\frac{4}{12} = \frac{1}{3}\)[/tex]
- [tex]\(\frac{4/3}{4} = \frac{1}{3}\)[/tex]
- [tex]\(\frac{4/9}{4/3} = \frac{1}{3}\)[/tex]
- The ratio between consecutive terms is constant [tex]\(\left(\frac{1}{3}\right)\)[/tex].
2. Since the ratio is constant, this sequence is Geometric.
#### Sequence 3: [tex]\(-4, -2, 1, 4, 8, \ldots\)[/tex]
1. Check if it is an Arithmetic Sequence:
- Check the differences between consecutive terms.
- [tex]\(-2 - (-4) = 2\)[/tex]
- [tex]\(1 - (-2) = 3\)[/tex]
- [tex]\(4 - 1 = 3\)[/tex]
- [tex]\(8 - 4 = 4\)[/tex]
- The differences between consecutive terms are not constant.
2. Check if it is a Geometric Sequence:
- Check the ratios between consecutive terms.
- [tex]\(\frac{-2}{-4} = \frac{1}{2}\)[/tex]
- [tex]\(\frac{1}{-2} = -\frac{1}{2}\)[/tex]
- [tex]\(\frac{4}{1} = 4\)[/tex]
- [tex]\(\frac{8}{4} = 2\)[/tex]
- The ratios between consecutive terms are not constant.
3. Since neither the differences nor the ratios are constant, this sequence is Neither.
### Classification:
- Arithmetic: [tex]\(-1, 3, 7, 11, 15, \ldots\)[/tex]
- Geometric: [tex]\(36, 12, 4, \frac{4}{3}, \frac{4}{9}, \ldots\)[/tex]
- Neither: [tex]\(-4, -2, 1, 4, 8, \ldots\)[/tex]
