Answer :
Sure! Let's go through each sequence step by step to determine whether they are arithmetic, geometric, or neither.
1. Sequence 1: 98.3, 94.1, 89.9, 85.7, ...
- This sequence involves finding the difference between terms to see if it's constant.
- Calculate the difference:
- [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 common difference is constant at [tex]\(-4.2\)[/tex], this is an arithmetic sequence.
2. Sequence 2: 1, 0, -1, 0, ...
- In an arithmetic sequence, there is a constant difference; in a geometric sequence, there is a constant ratio.
- This sequence alternates between 1, 0, and -1, back to 0.
- There is no constant difference or ratio present.
- Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence 3: 1.75, 3.5, 7, 14
- For a geometric sequence, each term is a constant multiple of the previous term.
- Calculate the ratio:
- [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- [tex]\( \frac{14}{7} = 2 \)[/tex]
- Since the common ratio is constant at [tex]\(2\)[/tex], this sequence is geometric.
4. Sequence 4: -12, -10.8, -9.6, -8.4
- Find the difference between terms:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- The common difference is [tex]\(1.2\)[/tex], which is constant.
- Therefore, this sequence is an arithmetic sequence.
5. Sequence 5: -1, 1, -1, 1, ...
- Check for a constant difference or ratio.
- This sequence alternates between -1 and 1.
- There's no constant difference or ratio.
- Hence, this sequence is neither arithmetic nor geometric.
Summarizing the sequences:
- Sequence 1 is arithmetic.
- Sequence 2 is neither.
- Sequence 3 is geometric.
- Sequence 4 is arithmetic.
- Sequence 5 is neither.
I hope this step-by-step explanation helps!
1. Sequence 1: 98.3, 94.1, 89.9, 85.7, ...
- This sequence involves finding the difference between terms to see if it's constant.
- Calculate the difference:
- [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 common difference is constant at [tex]\(-4.2\)[/tex], this is an arithmetic sequence.
2. Sequence 2: 1, 0, -1, 0, ...
- In an arithmetic sequence, there is a constant difference; in a geometric sequence, there is a constant ratio.
- This sequence alternates between 1, 0, and -1, back to 0.
- There is no constant difference or ratio present.
- Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence 3: 1.75, 3.5, 7, 14
- For a geometric sequence, each term is a constant multiple of the previous term.
- Calculate the ratio:
- [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- [tex]\( \frac{14}{7} = 2 \)[/tex]
- Since the common ratio is constant at [tex]\(2\)[/tex], this sequence is geometric.
4. Sequence 4: -12, -10.8, -9.6, -8.4
- Find the difference between terms:
- [tex]\(-10.8 - (-12) = 1.2\)[/tex]
- [tex]\(-9.6 - (-10.8) = 1.2\)[/tex]
- [tex]\(-8.4 - (-9.6) = 1.2\)[/tex]
- The common difference is [tex]\(1.2\)[/tex], which is constant.
- Therefore, this sequence is an arithmetic sequence.
5. Sequence 5: -1, 1, -1, 1, ...
- Check for a constant difference or ratio.
- This sequence alternates between -1 and 1.
- There's no constant difference or ratio.
- Hence, this sequence is neither arithmetic nor geometric.
Summarizing the sequences:
- Sequence 1 is arithmetic.
- Sequence 2 is neither.
- Sequence 3 is geometric.
- Sequence 4 is arithmetic.
- Sequence 5 is neither.
I hope this step-by-step explanation helps!
