Answer :
Sure! Let's determine whether each of the given sequences is arithmetic, geometric, or neither by examining their patterns.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if it's arithmetic, look for a common difference between consecutive terms.
- Subtract the first term from the second: [tex]\(94.1 - 98.3 = -4.2\)[/tex].
- Subtract the second term from the third: [tex]\(89.9 - 94.1 = -4.2\)[/tex].
- Since the differences are the same, this is an arithmetic sequence with a common difference of [tex]\(-4.2\)[/tex].
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check if it has a constant difference.
- Common differences found: [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], then [tex]\(0 - (-1) = 1\)[/tex].
- The differences alternate, so it doesn't have a constant difference.
- This sequence is neither arithmetic nor geometric. It appears to have a repeating pattern.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- To check if it's geometric, look for a common ratio between consecutive terms.
- Divide the second term by the first: [tex]\(3.5 / 1.75 = 2\)[/tex].
- Divide the third term by the second: [tex]\(7 / 3.5 = 2\)[/tex].
- Since the ratios are the same, this is a geometric sequence with a common ratio of [tex]\(2\)[/tex].
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check if it’s arithmetic by looking for a common difference.
- Calculate the difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex].
- Calculate the next difference: [tex]\(-9.6 - (-10.8) = 1.2\)[/tex].
- Since the differences are consistent, this is an arithmetic sequence with a common difference of [tex]\(1.2\)[/tex].
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Examine the differences: [tex]\(1 - (-1) = 2\)[/tex] and [tex]\(-1 - 1 = -2\)[/tex].
- The differences do not remain constant.
- This sequence is neither arithmetic nor geometric. It has a repeating pattern.
So, the classifications for the sequences are:
1. Arithmetic
2. Neither
3. Geometric
4. Arithmetic
5. Neither
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if it's arithmetic, look for a common difference between consecutive terms.
- Subtract the first term from the second: [tex]\(94.1 - 98.3 = -4.2\)[/tex].
- Subtract the second term from the third: [tex]\(89.9 - 94.1 = -4.2\)[/tex].
- Since the differences are the same, this is an arithmetic sequence with a common difference of [tex]\(-4.2\)[/tex].
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- Check if it has a constant difference.
- Common differences found: [tex]\(0 - 1 = -1\)[/tex], [tex]\(-1 - 0 = -1\)[/tex], then [tex]\(0 - (-1) = 1\)[/tex].
- The differences alternate, so it doesn't have a constant difference.
- This sequence is neither arithmetic nor geometric. It appears to have a repeating pattern.
3. Sequence: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- To check if it's geometric, look for a common ratio between consecutive terms.
- Divide the second term by the first: [tex]\(3.5 / 1.75 = 2\)[/tex].
- Divide the third term by the second: [tex]\(7 / 3.5 = 2\)[/tex].
- Since the ratios are the same, this is a geometric sequence with a common ratio of [tex]\(2\)[/tex].
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- Check if it’s arithmetic by looking for a common difference.
- Calculate the difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex].
- Calculate the next difference: [tex]\(-9.6 - (-10.8) = 1.2\)[/tex].
- Since the differences are consistent, this is an arithmetic sequence with a common difference of [tex]\(1.2\)[/tex].
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- Examine the differences: [tex]\(1 - (-1) = 2\)[/tex] and [tex]\(-1 - 1 = -2\)[/tex].
- The differences do not remain constant.
- This sequence is neither arithmetic nor geometric. It has a repeating pattern.
So, the classifications for the sequences are:
1. Arithmetic
2. Neither
3. Geometric
4. Arithmetic
5. Neither
