Answer :
We need to determine whether each sequence is arithmetic, geometric, or neither. Recall:
- A sequence is arithmetic if the difference between consecutive terms is constant. That is, if
[tex]$$a_{n+1} - a_n = d \quad \text{(constant for all } n\text{)}.$$[/tex]
- A sequence is geometric if the ratio between consecutive terms is constant. That is, if
[tex]$$\frac{a_{n+1}}{a_n} = r \quad \text{(constant for all } n\text{)},$$[/tex]
provided none of the terms used as a divisor are zero.
Let’s analyze each sequence step by step.
--------------------------------------------------
1. Sequence A:
[tex]$$98.3,\; 94.1,\; 89.9,\; 85.7,\ldots$$[/tex]
Step 1.1: Find the differences.
- First difference:
[tex]$$94.1 - 98.3 = -4.2.$$[/tex]
- Second difference:
[tex]$$89.9 - 94.1 = -4.2.$$[/tex]
- Third difference:
[tex]$$85.7 - 89.9 = -4.2.$$[/tex]
Conclusion:
Since all the differences are equal ([tex]$-4.2$[/tex]), the sequence is an arithmetic sequence.
--------------------------------------------------
2. Sequence B:
[tex]$$1,\; 0,\; -1,\; 0,\ldots$$[/tex]
Step 2.1: Check the differences.
- First difference:
[tex]$$0 - 1 = -1.$$[/tex]
- Second difference:
[tex]$$-1 - 0 = -1.$$[/tex]
- Third difference:
[tex]$$0 - (-1) = 1.$$[/tex]
The differences are not all the same since the third difference is [tex]$1$[/tex] instead of [tex]$-1$[/tex].
Step 2.2: Check the ratios (if possible).
- First ratio:
[tex]$$\frac{0}{1} = 0.$$[/tex]
- The next ratio would involve division by [tex]$0$[/tex] or otherwise be inconsistent since [tex]$0$[/tex] appears in the sequence.
Conclusion:
Since the differences are not constant and the ratios are not consistently defined, the sequence is neither arithmetic nor geometric.
--------------------------------------------------
3. Sequence C:
[tex]$$1.75,\; 3.5,\; 7,\; 14$$[/tex]
Step 3.1: Compute the ratios.
- First ratio:
[tex]$$\frac{3.5}{1.75} = 2.$$[/tex]
- Second ratio:
[tex]$$\frac{7}{3.5} = 2.$$[/tex]
- Third ratio:
[tex]$$\frac{14}{7} = 2.$$[/tex]
Conclusion:
Since the ratio is consistently [tex]$2$[/tex], this sequence is geometric.
--------------------------------------------------
4. Sequence D:
[tex]$$-12,\; -10.8,\; -9.6,\; -8.4$$[/tex]
Step 4.1: Determine the differences.
- First difference:
[tex]$$-10.8 - (-12) = 1.2.$$[/tex]
- Second difference:
[tex]$$-9.6 - (-10.8) = 1.2.$$[/tex]
- Third difference:
[tex]$$-8.4 - (-9.6) = 1.2.$$[/tex]
Conclusion:
Since the difference is constant ([tex]$1.2$[/tex]), the sequence is arithmetic.
--------------------------------------------------
5. Sequence E:
[tex]$$-1,\; 1,\; -1,\; 1,\ldots$$[/tex]
Step 5.1: Calculate the ratios.
- First ratio:
[tex]$$\frac{1}{-1} = -1.$$[/tex]
- Second ratio:
[tex]$$\frac{-1}{1} = -1.$$[/tex]
- Third ratio:
[tex]$$\frac{1}{-1} = -1.$$[/tex]
Conclusion:
Since the ratio is consistently [tex]$-1$[/tex], this sequence is geometric.
--------------------------------------------------
Final Classification of each sequence:
- [tex]$$98.3,\; 94.1,\; 89.9,\; 85.7,\ldots \quad \text{is Arithmetic.}$$[/tex]
- [tex]$$1,\; 0,\; -1,\; 0,\ldots \quad \text{is Neither.}$$[/tex]
- [tex]$$1.75,\; 3.5,\; 7,\; 14 \quad \text{is Geometric.}$$[/tex]
- [tex]$$-12,\; -10.8,\; -9.6,\; -8.4 \quad \text{is Arithmetic.}$$[/tex]
- [tex]$$-1,\; 1,\; -1,\; 1,\ldots \quad \text{is Geometric.}$$[/tex]
Each step verified that the successive differences or ratios are consistent, leading to our final classification for each sequence.
- A sequence is arithmetic if the difference between consecutive terms is constant. That is, if
[tex]$$a_{n+1} - a_n = d \quad \text{(constant for all } n\text{)}.$$[/tex]
- A sequence is geometric if the ratio between consecutive terms is constant. That is, if
[tex]$$\frac{a_{n+1}}{a_n} = r \quad \text{(constant for all } n\text{)},$$[/tex]
provided none of the terms used as a divisor are zero.
Let’s analyze each sequence step by step.
--------------------------------------------------
1. Sequence A:
[tex]$$98.3,\; 94.1,\; 89.9,\; 85.7,\ldots$$[/tex]
Step 1.1: Find the differences.
- First difference:
[tex]$$94.1 - 98.3 = -4.2.$$[/tex]
- Second difference:
[tex]$$89.9 - 94.1 = -4.2.$$[/tex]
- Third difference:
[tex]$$85.7 - 89.9 = -4.2.$$[/tex]
Conclusion:
Since all the differences are equal ([tex]$-4.2$[/tex]), the sequence is an arithmetic sequence.
--------------------------------------------------
2. Sequence B:
[tex]$$1,\; 0,\; -1,\; 0,\ldots$$[/tex]
Step 2.1: Check the differences.
- First difference:
[tex]$$0 - 1 = -1.$$[/tex]
- Second difference:
[tex]$$-1 - 0 = -1.$$[/tex]
- Third difference:
[tex]$$0 - (-1) = 1.$$[/tex]
The differences are not all the same since the third difference is [tex]$1$[/tex] instead of [tex]$-1$[/tex].
Step 2.2: Check the ratios (if possible).
- First ratio:
[tex]$$\frac{0}{1} = 0.$$[/tex]
- The next ratio would involve division by [tex]$0$[/tex] or otherwise be inconsistent since [tex]$0$[/tex] appears in the sequence.
Conclusion:
Since the differences are not constant and the ratios are not consistently defined, the sequence is neither arithmetic nor geometric.
--------------------------------------------------
3. Sequence C:
[tex]$$1.75,\; 3.5,\; 7,\; 14$$[/tex]
Step 3.1: Compute the ratios.
- First ratio:
[tex]$$\frac{3.5}{1.75} = 2.$$[/tex]
- Second ratio:
[tex]$$\frac{7}{3.5} = 2.$$[/tex]
- Third ratio:
[tex]$$\frac{14}{7} = 2.$$[/tex]
Conclusion:
Since the ratio is consistently [tex]$2$[/tex], this sequence is geometric.
--------------------------------------------------
4. Sequence D:
[tex]$$-12,\; -10.8,\; -9.6,\; -8.4$$[/tex]
Step 4.1: Determine the differences.
- First difference:
[tex]$$-10.8 - (-12) = 1.2.$$[/tex]
- Second difference:
[tex]$$-9.6 - (-10.8) = 1.2.$$[/tex]
- Third difference:
[tex]$$-8.4 - (-9.6) = 1.2.$$[/tex]
Conclusion:
Since the difference is constant ([tex]$1.2$[/tex]), the sequence is arithmetic.
--------------------------------------------------
5. Sequence E:
[tex]$$-1,\; 1,\; -1,\; 1,\ldots$$[/tex]
Step 5.1: Calculate the ratios.
- First ratio:
[tex]$$\frac{1}{-1} = -1.$$[/tex]
- Second ratio:
[tex]$$\frac{-1}{1} = -1.$$[/tex]
- Third ratio:
[tex]$$\frac{1}{-1} = -1.$$[/tex]
Conclusion:
Since the ratio is consistently [tex]$-1$[/tex], this sequence is geometric.
--------------------------------------------------
Final Classification of each sequence:
- [tex]$$98.3,\; 94.1,\; 89.9,\; 85.7,\ldots \quad \text{is Arithmetic.}$$[/tex]
- [tex]$$1,\; 0,\; -1,\; 0,\ldots \quad \text{is Neither.}$$[/tex]
- [tex]$$1.75,\; 3.5,\; 7,\; 14 \quad \text{is Geometric.}$$[/tex]
- [tex]$$-12,\; -10.8,\; -9.6,\; -8.4 \quad \text{is Arithmetic.}$$[/tex]
- [tex]$$-1,\; 1,\; -1,\; 1,\ldots \quad \text{is Geometric.}$$[/tex]
Each step verified that the successive differences or ratios are consistent, leading to our final classification for each sequence.
