Answer :
Sure, let's go through each sequence step by step and determine whether it is arithmetic, geometric, or neither.
### Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
To check if a sequence is arithmetic, we need to see if there is a constant difference between consecutive terms.
- Difference between [tex]\(94.1\)[/tex] and [tex]\(98.3\)[/tex]: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
- Difference between [tex]\(89.9\)[/tex] and [tex]\(94.1\)[/tex]: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
- Difference between [tex]\(85.7\)[/tex] and [tex]\(89.9\)[/tex]: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
Since the difference between consecutive terms is constant ([tex]\(-4.2\)[/tex]), this sequence is arithmetic.
### Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
To check if a sequence is arithmetic or geometric, we need to check for a constant difference or ratio.
- Difference between [tex]\(0\)[/tex] and [tex]\(1\)[/tex]: [tex]\( 0 - 1 = -1 \)[/tex]
- Difference between [tex]\(-1\)[/tex] and [tex]\(0\)[/tex]: [tex]\( -1 - 0 = -1 \)[/tex]
- Difference between [tex]\(0\)[/tex] and [tex]\(-1\)[/tex]: [tex]\( 0 - (-1) = 1 \)[/tex]
Since the difference is not constant and also there is no common ratio, this sequence is neither arithmetic nor geometric.
### Sequence 3: [tex]\(1.75, 35, 7, 14\)[/tex]
To check if a sequence is arithmetic or geometric, we need to check for a constant difference or ratio.
- Difference between [tex]\(35\)[/tex] and [tex]\(1.75\)[/tex]: [tex]\( 35 - 1.75 = 33.25 \)[/tex]
- Difference between [tex]\(7\)[/tex] and [tex]\(35\)[/tex]: [tex]\( 7 - 35 = -28 \)[/tex]
- Difference between [tex]\(14\)[/tex] and [tex]\(7\)[/tex]: [tex]\( 14 - 7 = 7 \)[/tex]
- Ratio between [tex]\(35\)[/tex] and [tex]\(1.75\)[/tex]: [tex]\( 35 / 1.75 = 20 \)[/tex]
- Ratio between [tex]\(7\)[/tex] and [tex]\(35\)[/tex]: [tex]\( 7 / 35 = 1/5 \)[/tex]
- Ratio between [tex]\(14\)[/tex] and [tex]\(7\)[/tex]: [tex]\( 14 / 7 = 2 \)[/tex]
Since neither the difference nor the ratio is consistent, this sequence is neither arithmetic nor geometric.
### Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
To check if a sequence is arithmetic, we need to see if there is a constant difference between consecutive terms.
- Difference between [tex]\(-10.8\)[/tex] and [tex]\(-12\)[/tex]: [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
- Difference between [tex]\(-9.6\)[/tex] and [tex]\(-10.8\)[/tex]: [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
- Difference between [tex]\(-8.4\)[/tex] and [tex]\(-9.6\)[/tex]: [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
Since the difference between consecutive terms is constant ([tex]\(1.2\)[/tex]), this sequence is arithmetic.
### Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
To check if a sequence is arithmetic or geometric, we need to check for a constant difference or ratio.
- Difference between [tex]\(1\)[/tex] and [tex]\(-1\)[/tex]: [tex]\( 1 - (-1) = 2 \)[/tex]
- Difference between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex]: [tex]\( -1 - 1 = -2 \)[/tex]
- Difference between [tex]\(1\)[/tex] and [tex]\(-1\)[/tex]: [tex]\( 1 - (-1) = 2 \)[/tex]
Since the difference is not constant and also there is no consistent ratio, this sequence is neither arithmetic nor geometric.
### Conclusion:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Neither
- Sequence 4: Arithmetic
- Sequence 5: Neither
So, the sequences are classified as follows:
[tex]\[
\begin{align*}
&98.3, 94.1, 89.9, 85.7, \ldots & \text{Arithmetic} \\
&1, 0, -1, 0, \ldots & \text{Neither} \\
&1.75, 35, 7, 14 & \text{Neither} \\
&-12, -10.8, -9.6, -8.4 & \text{Arithmetic} \\
&-1, 1, -1, 1, \ldots & \text{Neither} \\
\end{align*}
\][/tex]
### Sequence 1: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
To check if a sequence is arithmetic, we need to see if there is a constant difference between consecutive terms.
- Difference between [tex]\(94.1\)[/tex] and [tex]\(98.3\)[/tex]: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
- Difference between [tex]\(89.9\)[/tex] and [tex]\(94.1\)[/tex]: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
- Difference between [tex]\(85.7\)[/tex] and [tex]\(89.9\)[/tex]: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
Since the difference between consecutive terms is constant ([tex]\(-4.2\)[/tex]), this sequence is arithmetic.
### Sequence 2: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
To check if a sequence is arithmetic or geometric, we need to check for a constant difference or ratio.
- Difference between [tex]\(0\)[/tex] and [tex]\(1\)[/tex]: [tex]\( 0 - 1 = -1 \)[/tex]
- Difference between [tex]\(-1\)[/tex] and [tex]\(0\)[/tex]: [tex]\( -1 - 0 = -1 \)[/tex]
- Difference between [tex]\(0\)[/tex] and [tex]\(-1\)[/tex]: [tex]\( 0 - (-1) = 1 \)[/tex]
Since the difference is not constant and also there is no common ratio, this sequence is neither arithmetic nor geometric.
### Sequence 3: [tex]\(1.75, 35, 7, 14\)[/tex]
To check if a sequence is arithmetic or geometric, we need to check for a constant difference or ratio.
- Difference between [tex]\(35\)[/tex] and [tex]\(1.75\)[/tex]: [tex]\( 35 - 1.75 = 33.25 \)[/tex]
- Difference between [tex]\(7\)[/tex] and [tex]\(35\)[/tex]: [tex]\( 7 - 35 = -28 \)[/tex]
- Difference between [tex]\(14\)[/tex] and [tex]\(7\)[/tex]: [tex]\( 14 - 7 = 7 \)[/tex]
- Ratio between [tex]\(35\)[/tex] and [tex]\(1.75\)[/tex]: [tex]\( 35 / 1.75 = 20 \)[/tex]
- Ratio between [tex]\(7\)[/tex] and [tex]\(35\)[/tex]: [tex]\( 7 / 35 = 1/5 \)[/tex]
- Ratio between [tex]\(14\)[/tex] and [tex]\(7\)[/tex]: [tex]\( 14 / 7 = 2 \)[/tex]
Since neither the difference nor the ratio is consistent, this sequence is neither arithmetic nor geometric.
### Sequence 4: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
To check if a sequence is arithmetic, we need to see if there is a constant difference between consecutive terms.
- Difference between [tex]\(-10.8\)[/tex] and [tex]\(-12\)[/tex]: [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
- Difference between [tex]\(-9.6\)[/tex] and [tex]\(-10.8\)[/tex]: [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
- Difference between [tex]\(-8.4\)[/tex] and [tex]\(-9.6\)[/tex]: [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
Since the difference between consecutive terms is constant ([tex]\(1.2\)[/tex]), this sequence is arithmetic.
### Sequence 5: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
To check if a sequence is arithmetic or geometric, we need to check for a constant difference or ratio.
- Difference between [tex]\(1\)[/tex] and [tex]\(-1\)[/tex]: [tex]\( 1 - (-1) = 2 \)[/tex]
- Difference between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex]: [tex]\( -1 - 1 = -2 \)[/tex]
- Difference between [tex]\(1\)[/tex] and [tex]\(-1\)[/tex]: [tex]\( 1 - (-1) = 2 \)[/tex]
Since the difference is not constant and also there is no consistent ratio, this sequence is neither arithmetic nor geometric.
### Conclusion:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Neither
- Sequence 4: Arithmetic
- Sequence 5: Neither
So, the sequences are classified as follows:
[tex]\[
\begin{align*}
&98.3, 94.1, 89.9, 85.7, \ldots & \text{Arithmetic} \\
&1, 0, -1, 0, \ldots & \text{Neither} \\
&1.75, 35, 7, 14 & \text{Neither} \\
&-12, -10.8, -9.6, -8.4 & \text{Arithmetic} \\
&-1, 1, -1, 1, \ldots & \text{Neither} \\
\end{align*}
\][/tex]
