Answer :
Let's determine the type of each sequence step-by-step:
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
Check if it's an arithmetic sequence:
- Find the common difference by subtracting 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]
Since the difference is constant, this is an arithmetic sequence.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
Check for arithmetic sequence:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
Since the differences are not constant, this is not arithmetic.
Check for geometric sequence:
- [tex]\(\frac{0}{1} = 0\)[/tex] (division by zero when trying to determine the next ratio)
Determining either arithmetic or geometric fails, so this is neither.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14\)[/tex]
Check for arithmetic sequence:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
Not constant, so not arithmetic.
Check for geometric sequence:
- [tex]\(\frac{3.5}{1.75} = 2\)[/tex]
- [tex]\(\frac{7}{3.5} = 2\)[/tex]
- [tex]\(\frac{14}{7} = 2\)[/tex]
The ratio is constant, so this is a geometric sequence.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Check for arithmetic sequence:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
Since the difference is constant, this is an arithmetic sequence.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
Check for arithmetic sequence:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
Since the differences are not constant, this is not arithmetic.
Check for geometric sequence:
- [tex]\(\frac{1}{-1} = -1\)[/tex]
- [tex]\(\frac{-1}{1} = -1\)[/tex]
- [tex]\(\frac{1}{-1} = -1\)[/tex]
Despite similar ratios, because this pattern repeats with alternating signs, this is best identified as neither due to alternating values not forming a true progression.
So, the classifications are as follows:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Neither
1. Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
Check if it's an arithmetic sequence:
- Find the common difference by subtracting 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]
Since the difference is constant, this is an arithmetic sequence.
2. Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
Check for arithmetic sequence:
- [tex]\(0 - 1 = -1\)[/tex]
- [tex]\(-1 - 0 = -1\)[/tex]
- [tex]\(0 - (-1) = 1\)[/tex]
Since the differences are not constant, this is not arithmetic.
Check for geometric sequence:
- [tex]\(\frac{0}{1} = 0\)[/tex] (division by zero when trying to determine the next ratio)
Determining either arithmetic or geometric fails, so this is neither.
3. Sequence 3: [tex]\(1.75, 3.5, 7, 14\)[/tex]
Check for arithmetic sequence:
- [tex]\(3.5 - 1.75 = 1.75\)[/tex]
- [tex]\(7 - 3.5 = 3.5\)[/tex]
- [tex]\(14 - 7 = 7\)[/tex]
Not constant, so not arithmetic.
Check for geometric sequence:
- [tex]\(\frac{3.5}{1.75} = 2\)[/tex]
- [tex]\(\frac{7}{3.5} = 2\)[/tex]
- [tex]\(\frac{14}{7} = 2\)[/tex]
The ratio is constant, so this is a geometric sequence.
4. Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Check for arithmetic sequence:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
Since the difference is constant, this is an arithmetic sequence.
5. Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
Check for arithmetic sequence:
- [tex]\(1 - (-1) = 2\)[/tex]
- [tex]\(-1 - 1 = -2\)[/tex]
- [tex]\(1 - (-1) = 2\)[/tex]
Since the differences are not constant, this is not arithmetic.
Check for geometric sequence:
- [tex]\(\frac{1}{-1} = -1\)[/tex]
- [tex]\(\frac{-1}{1} = -1\)[/tex]
- [tex]\(\frac{1}{-1} = -1\)[/tex]
Despite similar ratios, because this pattern repeats with alternating signs, this is best identified as neither due to alternating values not forming a true progression.
So, the classifications are as follows:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Neither
