Answer :
Sure! Let's examine each sequence to determine whether it is arithmetic, geometric, or neither.
1. Sequence: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
- Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant.
- 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]
- Since the differences are all the same ([tex]\(-4.2\)[/tex]), this sequence is Arithmetic.
2. Sequence: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- This sequence does not have a consistent difference (arithmetic) or ratio (geometric).
- The terms oscillate between 1 and 0, suggesting a repeating pattern but not one fitting arithmetic or geometric definitions.
- Therefore, this sequence is Neither.
3. Sequence: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Geometric Sequence: In a geometric sequence, the ratio between consecutive terms is constant.
- Calculate the ratios:
- [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- [tex]\( \frac{14}{7} = 2 \)[/tex]
- Since the ratios are all the same (2), this sequence is Geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- 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]
- Since the differences are all the same (1.2), this sequence is Arithmetic.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- This sequence alternates between -1 and 1.
- It has neither a consistent difference nor a consistent ratio.
- This sequence is Neither.
In summary:
- [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] is Arithmetic.
- [tex]\( 1, 0, -1, 0, \ldots \)[/tex] is Neither.
- [tex]\( 1.75, 3.5, 7, 14 \)[/tex] is Geometric.
- [tex]\(-12, -10.8, -9.6, -8.4 \)[/tex] is Arithmetic.
- [tex]\(-1, 1, -1, 1, \ldots \)[/tex] is Neither.
1. Sequence: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
- Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant.
- 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]
- Since the differences are all the same ([tex]\(-4.2\)[/tex]), this sequence is Arithmetic.
2. Sequence: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- This sequence does not have a consistent difference (arithmetic) or ratio (geometric).
- The terms oscillate between 1 and 0, suggesting a repeating pattern but not one fitting arithmetic or geometric definitions.
- Therefore, this sequence is Neither.
3. Sequence: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Geometric Sequence: In a geometric sequence, the ratio between consecutive terms is constant.
- Calculate the ratios:
- [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- [tex]\( \frac{14}{7} = 2 \)[/tex]
- Since the ratios are all the same (2), this sequence is Geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4\)[/tex]
- 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]
- Since the differences are all the same (1.2), this sequence is Arithmetic.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- This sequence alternates between -1 and 1.
- It has neither a consistent difference nor a consistent ratio.
- This sequence is Neither.
In summary:
- [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] is Arithmetic.
- [tex]\( 1, 0, -1, 0, \ldots \)[/tex] is Neither.
- [tex]\( 1.75, 3.5, 7, 14 \)[/tex] is Geometric.
- [tex]\(-12, -10.8, -9.6, -8.4 \)[/tex] is Arithmetic.
- [tex]\(-1, 1, -1, 1, \ldots \)[/tex] is Neither.
