Answer :
Sure! Let's explore each sequence to determine if they are arithmetic, geometric, or neither.
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if a sequence is arithmetic, we see if there's a common difference between consecutive terms.
- Subtract the second term from the first: [tex]\(94.1 - 98.3 = -4.2\)[/tex].
- Subtract the third term from the second: [tex]\(89.9 - 94.1 = -4.2\)[/tex].
- Since the difference is consistent, this sequence is arithmetic.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- This sequence alternates between values and doesn't follow a consistent pattern in differences or ratios.
- The difference between consecutive terms alternates in sign.
- Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- To check for a geometric sequence, we find if there's a common ratio by dividing consecutive terms.
- Divide the second term by the first: [tex]\(3.5 / 1.75 = 2\)[/tex].
- Divide the third term by the second: [tex]\(7 / 3.5 = 2\)[/tex].
- The ratios are consistent, so this sequence is geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Check for a common difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex].
- Similarly, [tex]\(-9.6 - (-10.8) = 1.2\)[/tex].
- Since the difference is consistent, this sequence is arithmetic.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- The sequence alternates between -1 and 1.
- It doesn't have a consistent difference or ratio.
- Therefore, this sequence is neither arithmetic nor geometric.
To summarize:
- Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Geometric: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex] and [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
1. Sequence: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex]
- To check if a sequence is arithmetic, we see if there's a common difference between consecutive terms.
- Subtract the second term from the first: [tex]\(94.1 - 98.3 = -4.2\)[/tex].
- Subtract the third term from the second: [tex]\(89.9 - 94.1 = -4.2\)[/tex].
- Since the difference is consistent, this sequence is arithmetic.
2. Sequence: [tex]\(1, 0, -1, 0, \ldots\)[/tex]
- This sequence alternates between values and doesn't follow a consistent pattern in differences or ratios.
- The difference between consecutive terms alternates in sign.
- Therefore, this sequence is neither arithmetic nor geometric.
3. Sequence: [tex]\(1.75, 3.5, 7, 14, \ldots\)[/tex]
- To check for a geometric sequence, we find if there's a common ratio by dividing consecutive terms.
- Divide the second term by the first: [tex]\(3.5 / 1.75 = 2\)[/tex].
- Divide the third term by the second: [tex]\(7 / 3.5 = 2\)[/tex].
- The ratios are consistent, so this sequence is geometric.
4. Sequence: [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Check for a common difference: [tex]\(-10.8 - (-12) = 1.2\)[/tex].
- Similarly, [tex]\(-9.6 - (-10.8) = 1.2\)[/tex].
- Since the difference is consistent, this sequence is arithmetic.
5. Sequence: [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
- The sequence alternates between -1 and 1.
- It doesn't have a consistent difference or ratio.
- Therefore, this sequence is neither arithmetic nor geometric.
To summarize:
- Arithmetic: [tex]\(98.3, 94.1, 89.9, 85.7, \ldots\)[/tex] and [tex]\(-12, -10.8, -9.6, -8.4, \ldots\)[/tex]
- Geometric: [tex]\(1.75, 3.5, 7, 14\)[/tex]
- Neither: [tex]\(1, 0, -1, 0, \ldots\)[/tex] and [tex]\(-1, 1, -1, 1, \ldots\)[/tex]
