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Answer :
- Sequence 1 has non-constant differences and ratios, so it is neither arithmetic nor geometric.
- Sequence 2 has a constant ratio of -1, so it is geometric.
- Sequence 3 has a constant difference of 1.2, so it is arithmetic.
- Sequence 4 has a constant ratio of 2, so it is geometric.
- Sequence 5 has a constant difference of -4.2, so it is arithmetic.
- The classifications are: Sequence 1: Neither, Sequence 2: Geometric, Sequence 3: Arithmetic, Sequence 4: Geometric, Sequence 5: Arithmetic.
### Explanation
1. Understanding the Problem
We are given five sequences and need to classify each as arithmetic, geometric, or neither. An arithmetic sequence has a constant difference between consecutive terms, while a geometric sequence has a constant ratio between consecutive terms.
2. Analyzing Sequence 1
Sequence 1: $1, 0, -1, 0, \ldots$
The differences between consecutive terms are $0-1 = -1$, $-1-0 = -1$, $0-(-1) = 1$. Since the differences are not constant, the sequence is not arithmetic.
The ratios between consecutive terms are $0/1 = 0$, $-1/0$ (undefined). Since the ratios are not constant (and one is undefined), the sequence is not geometric. Therefore, the sequence is neither arithmetic nor geometric.
3. Analyzing Sequence 2
Sequence 2: $-1, 1, -1, 1, \ldots$
The differences between consecutive terms are $1 - (-1) = 2$, $-1 - 1 = -2$, $1 - (-1) = 2$. Since the differences are not constant, the sequence is not arithmetic.
The ratios between consecutive terms are $1/(-1) = -1$, $-1/1 = -1$, $1/(-1) = -1$. Since the ratios are constant, the sequence is geometric.
4. Analyzing Sequence 3
Sequence 3: $-12, -10.8, -9.6, -8.4$
The differences between consecutive terms are $-10.8 - (-12) = 1.2$, $-9.6 - (-10.8) = 1.2$, $-8.4 - (-9.6) = 1.2$. Since the differences are constant, the sequence is arithmetic.
5. Analyzing Sequence 4
Sequence 4: $1.75, 3.5, 7, 14$
The differences between consecutive terms are $3.5 - 1.75 = 1.75$, $7 - 3.5 = 3.5$, $14 - 7 = 7$. Since the differences are not constant, the sequence is not arithmetic.
The ratios between consecutive terms are $3.5/1.75 = 2$, $7/3.5 = 2$, $14/7 = 2$. Since the ratios are constant, the sequence is geometric.
6. Analyzing Sequence 5
Sequence 5: $98.3, 94.1, 89.9, 85.7, \ldots$
The differences between consecutive terms are $94.1 - 98.3 = -4.2$, $89.9 - 94.1 = -4.2$, $85.7 - 89.9 = -4.2$. Since the differences are constant, the sequence is arithmetic.
7. Final Classification
In summary:
Sequence 1: Neither
Sequence 2: Geometric
Sequence 3: Arithmetic
Sequence 4: Geometric
Sequence 5: Arithmetic
### Examples
Understanding sequences is crucial in finance for predicting investment growth. Arithmetic sequences model simple linear growth, like adding a fixed amount to savings each month. Geometric sequences, on the other hand, model exponential growth, such as compound interest on an investment. Recognizing these patterns helps in making informed financial decisions and forecasting future returns.
- Sequence 2 has a constant ratio of -1, so it is geometric.
- Sequence 3 has a constant difference of 1.2, so it is arithmetic.
- Sequence 4 has a constant ratio of 2, so it is geometric.
- Sequence 5 has a constant difference of -4.2, so it is arithmetic.
- The classifications are: Sequence 1: Neither, Sequence 2: Geometric, Sequence 3: Arithmetic, Sequence 4: Geometric, Sequence 5: Arithmetic.
### Explanation
1. Understanding the Problem
We are given five sequences and need to classify each as arithmetic, geometric, or neither. An arithmetic sequence has a constant difference between consecutive terms, while a geometric sequence has a constant ratio between consecutive terms.
2. Analyzing Sequence 1
Sequence 1: $1, 0, -1, 0, \ldots$
The differences between consecutive terms are $0-1 = -1$, $-1-0 = -1$, $0-(-1) = 1$. Since the differences are not constant, the sequence is not arithmetic.
The ratios between consecutive terms are $0/1 = 0$, $-1/0$ (undefined). Since the ratios are not constant (and one is undefined), the sequence is not geometric. Therefore, the sequence is neither arithmetic nor geometric.
3. Analyzing Sequence 2
Sequence 2: $-1, 1, -1, 1, \ldots$
The differences between consecutive terms are $1 - (-1) = 2$, $-1 - 1 = -2$, $1 - (-1) = 2$. Since the differences are not constant, the sequence is not arithmetic.
The ratios between consecutive terms are $1/(-1) = -1$, $-1/1 = -1$, $1/(-1) = -1$. Since the ratios are constant, the sequence is geometric.
4. Analyzing Sequence 3
Sequence 3: $-12, -10.8, -9.6, -8.4$
The differences between consecutive terms are $-10.8 - (-12) = 1.2$, $-9.6 - (-10.8) = 1.2$, $-8.4 - (-9.6) = 1.2$. Since the differences are constant, the sequence is arithmetic.
5. Analyzing Sequence 4
Sequence 4: $1.75, 3.5, 7, 14$
The differences between consecutive terms are $3.5 - 1.75 = 1.75$, $7 - 3.5 = 3.5$, $14 - 7 = 7$. Since the differences are not constant, the sequence is not arithmetic.
The ratios between consecutive terms are $3.5/1.75 = 2$, $7/3.5 = 2$, $14/7 = 2$. Since the ratios are constant, the sequence is geometric.
6. Analyzing Sequence 5
Sequence 5: $98.3, 94.1, 89.9, 85.7, \ldots$
The differences between consecutive terms are $94.1 - 98.3 = -4.2$, $89.9 - 94.1 = -4.2$, $85.7 - 89.9 = -4.2$. Since the differences are constant, the sequence is arithmetic.
7. Final Classification
In summary:
Sequence 1: Neither
Sequence 2: Geometric
Sequence 3: Arithmetic
Sequence 4: Geometric
Sequence 5: Arithmetic
### Examples
Understanding sequences is crucial in finance for predicting investment growth. Arithmetic sequences model simple linear growth, like adding a fixed amount to savings each month. Geometric sequences, on the other hand, model exponential growth, such as compound interest on an investment. Recognizing these patterns helps in making informed financial decisions and forecasting future returns.
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