Answer :
We are given five sequences:
1. [tex]\(98.3,\; 94.1,\; 89.9,\; 85.7,\ldots\)[/tex]
2. [tex]\(1,\; 0,\; -1,\; 0,\ldots\)[/tex]
3. [tex]\(1.75,\; 3.5,\; 7,\; 14\)[/tex]
4. [tex]\(-12,\; -10.8,\; -9.6,\; -8.4\)[/tex]
5. [tex]\(-1,\; 1,\; -1,\; 1,\ldots\)[/tex]
A sequence is termed:
- Arithmetic if the difference between consecutive terms is constant.
- Geometric if the ratio between consecutive terms is constant.
- Neither if it does not satisfy either condition.
Let’s examine each sequence step by step.
--------------------------------------------------
Sequence 1: [tex]\(98.3,\; 94.1,\; 89.9,\; 85.7,\ldots\)[/tex]
1. 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]
2. Since all differences are equal (common difference [tex]\(=-4.2\)[/tex]), this sequence is arithmetic.
--------------------------------------------------
Sequence 2: [tex]\(1,\; 0,\; -1,\; 0,\ldots\)[/tex]
1. Calculate the differences (to test for arithmetic behavior):
[tex]\[
0 - 1 = -1,
\][/tex]
[tex]\[
-1 - 0 = -1,
\][/tex]
[tex]\[
0 - (-1) = 1.
\][/tex]
2. The differences, [tex]\(-1, -1, 1\)[/tex], are not all equal. Therefore, it is not arithmetic.
3. Now, test for geometric behavior by considering ratios:
[tex]\[
\frac{0}{1} = 0.
\][/tex]
The next ratio would require division by zero (e.g., [tex]\(-1/0\)[/tex]), which is undefined. Thus, it is not geometric.
4. Therefore, Sequence 2 is neither arithmetic nor geometric.
--------------------------------------------------
Sequence 3: [tex]\(1.75,\; 3.5,\; 7,\; 14\)[/tex]
1. Test for arithmetic behavior:
[tex]\[
3.5 - 1.75 = 1.75,
\][/tex]
[tex]\[
7 - 3.5 = 3.5,
\][/tex]
[tex]\[
14 - 7 = 7.
\][/tex]
The differences are not constant; so, it is not arithmetic.
2. Test for geometric behavior by finding the ratios:
[tex]\[
\frac{3.5}{1.75} = 2,
\][/tex]
[tex]\[
\frac{7}{3.5} = 2,
\][/tex]
[tex]\[
\frac{14}{7} = 2.
\][/tex]
The constant ratio is [tex]\(2\)[/tex]. Therefore, Sequence 3 is geometric.
--------------------------------------------------
Sequence 4: [tex]\(-12,\; -10.8,\; -9.6,\; -8.4\)[/tex]
1. Calculate the differences:
[tex]\[
-10.8 - (-12) = 1.2,
\][/tex]
[tex]\[
-9.6 - (-10.8) = 1.2,
\][/tex]
[tex]\[
-8.4 - (-9.6) = 1.2.
\][/tex]
2. With a constant difference of [tex]\(1.2\)[/tex], Sequence 4 is arithmetic.
--------------------------------------------------
Sequence 5: [tex]\(-1,\; 1,\; -1,\; 1,\ldots\)[/tex]
1. Test for arithmetic behavior:
[tex]\[
1 - (-1) = 2,
\][/tex]
[tex]\[
-1 - 1 = -2,
\][/tex]
The differences [tex]\(2\)[/tex] and [tex]\(-2\)[/tex] are not the same; therefore, it is not arithmetic.
2. Test for geometric behavior by considering ratios:
[tex]\[
\frac{1}{-1} = -1,
\][/tex]
[tex]\[
\frac{-1}{1} = -1.
\][/tex]
The constant ratio is [tex]\(-1\)[/tex]. Therefore, Sequence 5 is geometric.
--------------------------------------------------
Now, we summarize our classifications:
- Arithmetic Sequences:
Sequence 1: [tex]\(98.3,\; 94.1,\; 89.9,\; 85.7,\ldots\)[/tex] (common difference [tex]\(-4.2\)[/tex])
Sequence 4: [tex]\(-12,\; -10.8,\; -9.6,\; -8.4\)[/tex] (common difference [tex]\(1.2\)[/tex])
- Geometric Sequences:
Sequence 3: [tex]\(1.75,\; 3.5,\; 7,\; 14\)[/tex] (common ratio [tex]\(2\)[/tex])
Sequence 5: [tex]\(-1,\; 1,\; -1,\; 1,\ldots\)[/tex] (common ratio [tex]\(-1\)[/tex])
- Neither:
Sequence 2: [tex]\(1,\; 0,\; -1,\; 0,\ldots\)[/tex]
Thus, the final answer is:
Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Geometric: [tex]\(1.75, 3.5, 7, 14\)[/tex] and [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
1. [tex]\(98.3,\; 94.1,\; 89.9,\; 85.7,\ldots\)[/tex]
2. [tex]\(1,\; 0,\; -1,\; 0,\ldots\)[/tex]
3. [tex]\(1.75,\; 3.5,\; 7,\; 14\)[/tex]
4. [tex]\(-12,\; -10.8,\; -9.6,\; -8.4\)[/tex]
5. [tex]\(-1,\; 1,\; -1,\; 1,\ldots\)[/tex]
A sequence is termed:
- Arithmetic if the difference between consecutive terms is constant.
- Geometric if the ratio between consecutive terms is constant.
- Neither if it does not satisfy either condition.
Let’s examine each sequence step by step.
--------------------------------------------------
Sequence 1: [tex]\(98.3,\; 94.1,\; 89.9,\; 85.7,\ldots\)[/tex]
1. 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]
2. Since all differences are equal (common difference [tex]\(=-4.2\)[/tex]), this sequence is arithmetic.
--------------------------------------------------
Sequence 2: [tex]\(1,\; 0,\; -1,\; 0,\ldots\)[/tex]
1. Calculate the differences (to test for arithmetic behavior):
[tex]\[
0 - 1 = -1,
\][/tex]
[tex]\[
-1 - 0 = -1,
\][/tex]
[tex]\[
0 - (-1) = 1.
\][/tex]
2. The differences, [tex]\(-1, -1, 1\)[/tex], are not all equal. Therefore, it is not arithmetic.
3. Now, test for geometric behavior by considering ratios:
[tex]\[
\frac{0}{1} = 0.
\][/tex]
The next ratio would require division by zero (e.g., [tex]\(-1/0\)[/tex]), which is undefined. Thus, it is not geometric.
4. Therefore, Sequence 2 is neither arithmetic nor geometric.
--------------------------------------------------
Sequence 3: [tex]\(1.75,\; 3.5,\; 7,\; 14\)[/tex]
1. Test for arithmetic behavior:
[tex]\[
3.5 - 1.75 = 1.75,
\][/tex]
[tex]\[
7 - 3.5 = 3.5,
\][/tex]
[tex]\[
14 - 7 = 7.
\][/tex]
The differences are not constant; so, it is not arithmetic.
2. Test for geometric behavior by finding the ratios:
[tex]\[
\frac{3.5}{1.75} = 2,
\][/tex]
[tex]\[
\frac{7}{3.5} = 2,
\][/tex]
[tex]\[
\frac{14}{7} = 2.
\][/tex]
The constant ratio is [tex]\(2\)[/tex]. Therefore, Sequence 3 is geometric.
--------------------------------------------------
Sequence 4: [tex]\(-12,\; -10.8,\; -9.6,\; -8.4\)[/tex]
1. Calculate the differences:
[tex]\[
-10.8 - (-12) = 1.2,
\][/tex]
[tex]\[
-9.6 - (-10.8) = 1.2,
\][/tex]
[tex]\[
-8.4 - (-9.6) = 1.2.
\][/tex]
2. With a constant difference of [tex]\(1.2\)[/tex], Sequence 4 is arithmetic.
--------------------------------------------------
Sequence 5: [tex]\(-1,\; 1,\; -1,\; 1,\ldots\)[/tex]
1. Test for arithmetic behavior:
[tex]\[
1 - (-1) = 2,
\][/tex]
[tex]\[
-1 - 1 = -2,
\][/tex]
The differences [tex]\(2\)[/tex] and [tex]\(-2\)[/tex] are not the same; therefore, it is not arithmetic.
2. Test for geometric behavior by considering ratios:
[tex]\[
\frac{1}{-1} = -1,
\][/tex]
[tex]\[
\frac{-1}{1} = -1.
\][/tex]
The constant ratio is [tex]\(-1\)[/tex]. Therefore, Sequence 5 is geometric.
--------------------------------------------------
Now, we summarize our classifications:
- Arithmetic Sequences:
Sequence 1: [tex]\(98.3,\; 94.1,\; 89.9,\; 85.7,\ldots\)[/tex] (common difference [tex]\(-4.2\)[/tex])
Sequence 4: [tex]\(-12,\; -10.8,\; -9.6,\; -8.4\)[/tex] (common difference [tex]\(1.2\)[/tex])
- Geometric Sequences:
Sequence 3: [tex]\(1.75,\; 3.5,\; 7,\; 14\)[/tex] (common ratio [tex]\(2\)[/tex])
Sequence 5: [tex]\(-1,\; 1,\; -1,\; 1,\ldots\)[/tex] (common ratio [tex]\(-1\)[/tex])
- Neither:
Sequence 2: [tex]\(1,\; 0,\; -1,\; 0,\ldots\)[/tex]
Thus, the final answer is:
Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
Geometric: [tex]\(1.75, 3.5, 7, 14\)[/tex] and [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
