Answer :
Sure, let's analyze each of the given sequences to determine whether they are arithmetic, geometric, or neither.
1. [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
To determine if a sequence is arithmetic, we check if there is a constant difference between consecutive terms.
Difference between 94.1 and 98.3: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
Difference between 89.9 and 94.1: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
Difference between 85.7 and 89.9: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
Since the difference is constant ([tex]\(-4.2\)[/tex]), this sequence is arithmetic.
2. [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
Let's check for a constant difference first:
Difference between 0 and 1: [tex]\( 0 - 1 = -1 \)[/tex]
Difference between -1 and 0: [tex]\( -1 - 0 = -1 \)[/tex]
Difference between 0 and -1: [tex]\( 0 - (-1) = 1 \)[/tex]
The differences are not consistent.
Next, let's check for a constant ratio:
Ratio of 0 to 1: [tex]\( 0 / 1 = 0 \)[/tex]
Ratio of -1 to 0 is undefined because division by zero is not allowed.
As neither the difference nor the ratio is constant, this sequence is neither arithmetic nor geometric.
3. [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
To determine if a sequence is geometric, we check if there is a constant ratio between consecutive terms.
Ratio between 3.5 and 1.75: [tex]\( 3.5 / 1.75 = 2 \)[/tex]
Ratio between 7 and 3.5: [tex]\( 7 / 3.5 = 2 \)[/tex]
Ratio between 14 and 7: [tex]\( 14 / 7 = 2 \)[/tex]
Since the ratio is constant (2), this sequence is geometric.
4. [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
Let's check for a constant difference first:
Difference between -10.8 and -12: [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
Difference between -9.6 and -10.8: [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
Difference between -8.4 and -9.6: [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
Since the difference is constant (1.2), this sequence is arithmetic.
5. [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
Let’s first check for a constant difference:
Difference between 1 and -1: [tex]\( 1 - (-1) = 2 \)[/tex]
Difference between -1 and 1: [tex]\( -1 - 1 = -2 \)[/tex]
Difference between 1 and -1: [tex]\( 1 - (-1) = 2 \)[/tex]
The differences alternate between 2 and -2, which means they are not consistent.
Next, let’s check for a constant ratio:
Ratio between 1 and -1: [tex]\( 1 / (-1) = -1 \)[/tex]
Ratio between -1 and 1: [tex]\( -1 / 1 = -1 \)[/tex]
Ratio between 1 and -1: [tex]\( 1 / (-1) = -1 \)[/tex]
Although the ratio appears to be constant ([tex]\(-1\)[/tex]), a geometric sequence with a ratio of -1 would alternate its sign every term consistently. However, this sequence follows a pattern but does not strictly belong to a common ratio-based sequence.
Thus, this sequence is considered neither arithmetic nor geometric.
### Summary:
1. [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] - Arithmetic
2. [tex]\( 1, 0, -1, 0, \ldots \)[/tex] - Neither
3. [tex]\( 1.75, 3.5, 7, 14 \)[/tex] - Geometric
4. [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex] - Arithmetic
5. [tex]\( -1, 1, -1, 1, \ldots \)[/tex] - Neither
1. [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
To determine if a sequence is arithmetic, we check if there is a constant difference between consecutive terms.
Difference between 94.1 and 98.3: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
Difference between 89.9 and 94.1: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
Difference between 85.7 and 89.9: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
Since the difference is constant ([tex]\(-4.2\)[/tex]), this sequence is arithmetic.
2. [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
Let's check for a constant difference first:
Difference between 0 and 1: [tex]\( 0 - 1 = -1 \)[/tex]
Difference between -1 and 0: [tex]\( -1 - 0 = -1 \)[/tex]
Difference between 0 and -1: [tex]\( 0 - (-1) = 1 \)[/tex]
The differences are not consistent.
Next, let's check for a constant ratio:
Ratio of 0 to 1: [tex]\( 0 / 1 = 0 \)[/tex]
Ratio of -1 to 0 is undefined because division by zero is not allowed.
As neither the difference nor the ratio is constant, this sequence is neither arithmetic nor geometric.
3. [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
To determine if a sequence is geometric, we check if there is a constant ratio between consecutive terms.
Ratio between 3.5 and 1.75: [tex]\( 3.5 / 1.75 = 2 \)[/tex]
Ratio between 7 and 3.5: [tex]\( 7 / 3.5 = 2 \)[/tex]
Ratio between 14 and 7: [tex]\( 14 / 7 = 2 \)[/tex]
Since the ratio is constant (2), this sequence is geometric.
4. [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
Let's check for a constant difference first:
Difference between -10.8 and -12: [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
Difference between -9.6 and -10.8: [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
Difference between -8.4 and -9.6: [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
Since the difference is constant (1.2), this sequence is arithmetic.
5. [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
Let’s first check for a constant difference:
Difference between 1 and -1: [tex]\( 1 - (-1) = 2 \)[/tex]
Difference between -1 and 1: [tex]\( -1 - 1 = -2 \)[/tex]
Difference between 1 and -1: [tex]\( 1 - (-1) = 2 \)[/tex]
The differences alternate between 2 and -2, which means they are not consistent.
Next, let’s check for a constant ratio:
Ratio between 1 and -1: [tex]\( 1 / (-1) = -1 \)[/tex]
Ratio between -1 and 1: [tex]\( -1 / 1 = -1 \)[/tex]
Ratio between 1 and -1: [tex]\( 1 / (-1) = -1 \)[/tex]
Although the ratio appears to be constant ([tex]\(-1\)[/tex]), a geometric sequence with a ratio of -1 would alternate its sign every term consistently. However, this sequence follows a pattern but does not strictly belong to a common ratio-based sequence.
Thus, this sequence is considered neither arithmetic nor geometric.
### Summary:
1. [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex] - Arithmetic
2. [tex]\( 1, 0, -1, 0, \ldots \)[/tex] - Neither
3. [tex]\( 1.75, 3.5, 7, 14 \)[/tex] - Geometric
4. [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex] - Arithmetic
5. [tex]\( -1, 1, -1, 1, \ldots \)[/tex] - Neither
