Answer :
- Sequence 1 has non-constant differences and ratios, thus it is neither arithmetic nor geometric.
- Sequence 2 has a constant ratio of 2, thus it is geometric.
- Sequence 3 has a constant difference of 1.2, thus it is arithmetic.
- Sequence 4 has a constant ratio of -1, thus it is geometric.
- Final Answer:
Sequence 1: Neither
Sequence 2: Geometric
Sequence 3: Arithmetic
Sequence 4: Geometric
### Explanation
1. Understanding the Problem
We are given four sequences and we need to classify each as arithmetic, geometric, or neither. A sequence is arithmetic if the difference between consecutive terms is constant, and geometric if the ratio between consecutive terms is constant.
2. Analyzing Sequence 1
Sequence 1: $98.3, 94.1, 98.9, 85.7, \ldots$
The differences between consecutive terms are: $94.1 - 98.3 = -4.2$, $98.9 - 94.1 = 4.8$, $85.7 - 98.9 = -13.2$. The differences are not constant, so the sequence is not arithmetic.
The ratios between consecutive terms are: $\frac{94.1}{98.3} \approx 0.957$, $\frac{98.9}{94.1} \approx 1.051$, $\frac{85.7}{98.9} \approx 0.867$. The ratios are not constant, so the sequence is not geometric.
Therefore, the first sequence is neither arithmetic nor geometric.
3. Analyzing Sequence 2
Sequence 2: $1.75, 3.5, 7, 14, \ldots$
The differences between consecutive terms are: $3.5 - 1.75 = 1.75$, $7 - 3.5 = 3.5$, $14 - 7 = 7$. The differences are not constant, so the sequence is not arithmetic.
The ratios between consecutive terms are: $\frac{3.5}{1.75} = 2$, $\frac{7}{3.5} = 2$, $\frac{14}{7} = 2$. The ratios are constant, so the sequence is geometric.
Therefore, the second sequence is geometric.
4. Analyzing Sequence 3
Sequence 3: $-12, -10.8, -9.6, -8.4, \ldots$
The differences between consecutive terms are: $-10.8 - (-12) = 1.2$, $-9.6 - (-10.8) = 1.2$, $-8.4 - (-9.6) = 1.2$. The differences are constant, so the sequence is arithmetic.
The ratios between consecutive terms are: $\frac{-10.8}{-12} = 0.9$, $\frac{-9.6}{-10.8} \approx 0.889$, $\frac{-8.4}{-9.6} = 0.875$. The ratios are not constant, so the sequence is not geometric.
Therefore, the third sequence is arithmetic.
5. Analyzing Sequence 4
Sequence 4: $-1, 1, -1, 1, \ldots$
The differences between consecutive terms are: $1 - (-1) = 2$, $-1 - 1 = -2$, $1 - (-1) = 2$. The differences are not constant, so the sequence is not arithmetic.
The ratios between consecutive terms are: $\frac{1}{-1} = -1$, $\frac{-1}{1} = -1$, $\frac{1}{-1} = -1$. The ratios are constant, so the sequence is geometric.
Therefore, the fourth sequence is geometric.
6. Final Classification
In summary:
Sequence 1: Neither
Sequence 2: Geometric
Sequence 3: Arithmetic
Sequence 4: Geometric
### Examples
Understanding sequences is crucial in many real-world applications, such as predicting population growth, analyzing financial investments, or even designing efficient algorithms. For instance, geometric sequences can model compound interest, where the amount grows by a fixed percentage each year. Arithmetic sequences can be used to predict the cost of a project that increases by a fixed amount each month. Recognizing these patterns helps in making informed decisions and accurate forecasts.
- Sequence 2 has a constant ratio of 2, thus it is geometric.
- Sequence 3 has a constant difference of 1.2, thus it is arithmetic.
- Sequence 4 has a constant ratio of -1, thus it is geometric.
- Final Answer:
Sequence 1: Neither
Sequence 2: Geometric
Sequence 3: Arithmetic
Sequence 4: Geometric
### Explanation
1. Understanding the Problem
We are given four sequences and we need to classify each as arithmetic, geometric, or neither. A sequence is arithmetic if the difference between consecutive terms is constant, and geometric if the ratio between consecutive terms is constant.
2. Analyzing Sequence 1
Sequence 1: $98.3, 94.1, 98.9, 85.7, \ldots$
The differences between consecutive terms are: $94.1 - 98.3 = -4.2$, $98.9 - 94.1 = 4.8$, $85.7 - 98.9 = -13.2$. The differences are not constant, so the sequence is not arithmetic.
The ratios between consecutive terms are: $\frac{94.1}{98.3} \approx 0.957$, $\frac{98.9}{94.1} \approx 1.051$, $\frac{85.7}{98.9} \approx 0.867$. The ratios are not constant, so the sequence is not geometric.
Therefore, the first sequence is neither arithmetic nor geometric.
3. Analyzing Sequence 2
Sequence 2: $1.75, 3.5, 7, 14, \ldots$
The differences between consecutive terms are: $3.5 - 1.75 = 1.75$, $7 - 3.5 = 3.5$, $14 - 7 = 7$. The differences are not constant, so the sequence is not arithmetic.
The ratios between consecutive terms are: $\frac{3.5}{1.75} = 2$, $\frac{7}{3.5} = 2$, $\frac{14}{7} = 2$. The ratios are constant, so the sequence is geometric.
Therefore, the second sequence is geometric.
4. Analyzing Sequence 3
Sequence 3: $-12, -10.8, -9.6, -8.4, \ldots$
The differences between consecutive terms are: $-10.8 - (-12) = 1.2$, $-9.6 - (-10.8) = 1.2$, $-8.4 - (-9.6) = 1.2$. The differences are constant, so the sequence is arithmetic.
The ratios between consecutive terms are: $\frac{-10.8}{-12} = 0.9$, $\frac{-9.6}{-10.8} \approx 0.889$, $\frac{-8.4}{-9.6} = 0.875$. The ratios are not constant, so the sequence is not geometric.
Therefore, the third sequence is arithmetic.
5. Analyzing Sequence 4
Sequence 4: $-1, 1, -1, 1, \ldots$
The differences between consecutive terms are: $1 - (-1) = 2$, $-1 - 1 = -2$, $1 - (-1) = 2$. The differences are not constant, so the sequence is not arithmetic.
The ratios between consecutive terms are: $\frac{1}{-1} = -1$, $\frac{-1}{1} = -1$, $\frac{1}{-1} = -1$. The ratios are constant, so the sequence is geometric.
Therefore, the fourth sequence is geometric.
6. Final Classification
In summary:
Sequence 1: Neither
Sequence 2: Geometric
Sequence 3: Arithmetic
Sequence 4: Geometric
### Examples
Understanding sequences is crucial in many real-world applications, such as predicting population growth, analyzing financial investments, or even designing efficient algorithms. For instance, geometric sequences can model compound interest, where the amount grows by a fixed percentage each year. Arithmetic sequences can be used to predict the cost of a project that increases by a fixed amount each month. Recognizing these patterns helps in making informed decisions and accurate forecasts.
