Answer :
Sure! Let's examine each sequence and determine whether it's arithmetic, geometric, or neither based on their patterns.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To determine if a sequence is arithmetic, we check whether the difference between consecutive terms is constant.
- Calculate the differences:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- Since all the differences are the same, the sequence is arithmetic.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check if there's a constant difference or ratio.
- Differences vary:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences are not consistent, and the sequence doesn't follow a constant ratio. It's neither arithmetic nor geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14 \ldots\)[/tex]
- To determine if it's geometric, check for a constant ratio.
- Ratios:
- [tex]\(3.5 \div 1.75 = 2\)[/tex]
- [tex]\(7 \div 3.5 = 2\)[/tex]
- [tex]\(14 \div 7 = 2\)[/tex]
- All ratios are the same, confirming this is a geometric sequence.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check for a constant difference.
- Differences:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- All differences are equal, so this sequence is arithmetic.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Look for consistent differences or ratios.
- Differences alternate:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- The pattern alternates and doesn't maintain a single consistent difference or ratio. It is neither arithmetic nor geometric.
In conclusion, the characteristics of each sequence are as follows:
- 1st sequence: Arithmetic
- 2nd sequence: Neither
- 3rd sequence: Geometric
- 4th sequence: Arithmetic
- 5th sequence: Neither
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To determine if a sequence is arithmetic, we check whether the difference between consecutive terms is constant.
- Calculate the differences:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- Since all the differences are the same, the sequence is arithmetic.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check if there's a constant difference or ratio.
- Differences vary:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
- The differences are not consistent, and the sequence doesn't follow a constant ratio. It's neither arithmetic nor geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14 \ldots\)[/tex]
- To determine if it's geometric, check for a constant ratio.
- Ratios:
- [tex]\(3.5 \div 1.75 = 2\)[/tex]
- [tex]\(7 \div 3.5 = 2\)[/tex]
- [tex]\(14 \div 7 = 2\)[/tex]
- All ratios are the same, confirming this is a geometric sequence.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check for a constant difference.
- Differences:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- All differences are equal, so this sequence is arithmetic.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Look for consistent differences or ratios.
- Differences alternate:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
- The pattern alternates and doesn't maintain a single consistent difference or ratio. It is neither arithmetic nor geometric.
In conclusion, the characteristics of each sequence are as follows:
- 1st sequence: Arithmetic
- 2nd sequence: Neither
- 3rd sequence: Geometric
- 4th sequence: Arithmetic
- 5th sequence: Neither
