Answer :
To determine if each sequence is arithmetic, geometric, or neither, we need to check the properties of arithmetic and geometric sequences.
Arithmetic Sequence:
- In an arithmetic sequence, the difference between consecutive terms is constant. This common difference can be found by subtracting any term from the subsequent term.
Geometric Sequence:
- In a geometric sequence, the ratio between consecutive terms is constant. This common ratio can be found by dividing any term by the previous term.
Let's evaluate each sequence:
1. [tex]$98.3, 94.1, 89.9, 85.7$[/tex]
- Calculate the difference between consecutive terms:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- The difference is constant, so this sequence is Arithmetic.
2. [tex]$1, 0, -1, 0, \ldots$[/tex]
- Calculate the differences and check for a constant ratio:
- [tex]\(0 - 1 = -1\)[/tex] and [tex]\((-1) - 0 = -1\)[/tex], but then [tex]\(0 - (-1) = 1\)[/tex]
- Division for ratio: [tex]\(0/1 = 0\)[/tex], [tex]\((-1)/0\)[/tex] is undefined
- This sequence does not have a constant difference or ratio. So, it is Neither.
3. [tex]$1.75, 3.5, 7, 14$[/tex]
- Check the ratio between consecutive terms:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The ratio is constant, so this sequence is Geometric.
4. [tex]$-12, -10.8, -9.6, -8.4$[/tex]
- Calculate the difference between consecutive terms:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- The difference is constant, so this sequence is Arithmetic.
5. [tex]$-1, 1, -1, 1, \ldots$[/tex]
- Calculate the differences and check for a constant ratio:
- [tex]\(1 - (-1) = 2\)[/tex] and [tex]\((-1) - 1 = -2\)[/tex]
- Division for ratio: [tex]\(1/(-1) = -1\)[/tex], [tex]\((-1)/1 = -1\)[/tex]
- The differences are not consistent, and while the ratio alternates, it doesn't form a consistent pattern either. This sequence is Neither.
Summary:
- [tex]$98.3, 94.1, 89.9, 85.7$[/tex]: Arithmetic
- [tex]$1, 0, -1, 0, \ldots$[/tex]: Neither
- [tex]$1.75, 3.5, 7, 14$[/tex]: Geometric
- [tex]$-12, -10.8, -9.6, -8.4$[/tex]: Arithmetic
- [tex]$-1, 1, -1, 1, \ldots$[/tex]: Neither
Arithmetic Sequence:
- In an arithmetic sequence, the difference between consecutive terms is constant. This common difference can be found by subtracting any term from the subsequent term.
Geometric Sequence:
- In a geometric sequence, the ratio between consecutive terms is constant. This common ratio can be found by dividing any term by the previous term.
Let's evaluate each sequence:
1. [tex]$98.3, 94.1, 89.9, 85.7$[/tex]
- Calculate the difference between consecutive terms:
- [tex]\(94.1 - 98.3 = -4.2\)[/tex]
- [tex]\(89.9 - 94.1 = -4.2\)[/tex]
- [tex]\(85.7 - 89.9 = -4.2\)[/tex]
- The difference is constant, so this sequence is Arithmetic.
2. [tex]$1, 0, -1, 0, \ldots$[/tex]
- Calculate the differences and check for a constant ratio:
- [tex]\(0 - 1 = -1\)[/tex] and [tex]\((-1) - 0 = -1\)[/tex], but then [tex]\(0 - (-1) = 1\)[/tex]
- Division for ratio: [tex]\(0/1 = 0\)[/tex], [tex]\((-1)/0\)[/tex] is undefined
- This sequence does not have a constant difference or ratio. So, it is Neither.
3. [tex]$1.75, 3.5, 7, 14$[/tex]
- Check the ratio between consecutive terms:
- [tex]\(3.5 / 1.75 = 2\)[/tex]
- [tex]\(7 / 3.5 = 2\)[/tex]
- [tex]\(14 / 7 = 2\)[/tex]
- The ratio is constant, so this sequence is Geometric.
4. [tex]$-12, -10.8, -9.6, -8.4$[/tex]
- Calculate the difference between consecutive terms:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- The difference is constant, so this sequence is Arithmetic.
5. [tex]$-1, 1, -1, 1, \ldots$[/tex]
- Calculate the differences and check for a constant ratio:
- [tex]\(1 - (-1) = 2\)[/tex] and [tex]\((-1) - 1 = -2\)[/tex]
- Division for ratio: [tex]\(1/(-1) = -1\)[/tex], [tex]\((-1)/1 = -1\)[/tex]
- The differences are not consistent, and while the ratio alternates, it doesn't form a consistent pattern either. This sequence is Neither.
Summary:
- [tex]$98.3, 94.1, 89.9, 85.7$[/tex]: Arithmetic
- [tex]$1, 0, -1, 0, \ldots$[/tex]: Neither
- [tex]$1.75, 3.5, 7, 14$[/tex]: Geometric
- [tex]$-12, -10.8, -9.6, -8.4$[/tex]: Arithmetic
- [tex]$-1, 1, -1, 1, \ldots$[/tex]: Neither
