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Answer :
Sure! Let's go through each sequence step-by-step to determine if it is arithmetic, geometric, or neither.
1. Sequence 1: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
- To check if a sequence is arithmetic, we find the common difference by subtracting the first term from the second, the second term from the third, and so on.
- [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
- [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
- [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
- Each difference is the same, [tex]\(-4.2\)[/tex]. Hence, this sequence is Arithmetic.
2. Sequence 2: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- Let's check if it's arithmetic or geometric.
- For arithmetic, we calculate differences:
- [tex]\( 0 - 1 = -1 \)[/tex]
- [tex]\( -1 - 0 = -1 \)[/tex] (this seems incorrect as -1 - 0 = -1)
- [tex]\( 0 - (-1) = 1 \)[/tex]
- The differences are not consistent, so it is not an arithmetic sequence.
- Now for a geometric sequence, we check the ratios:
- [tex]\( \frac{0}{1} = 0 \)[/tex] (undefined ratio due to division by zero)
- The ratios aren't consistent either.
- This sequence is alternating, not fitting arithmetic or geometric patterns. Hence, it is Neither.
3. Sequence 3: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Check the differences for arithmetic:
- [tex]\( 3.5 - 1.75 = 1.75 \)[/tex]
- [tex]\( 7 - 3.5 = 3.5 \)[/tex]
- [tex]\( 14 - 7 = 7 \)[/tex]
- The differences aren't consistent, so it is not arithmetic.
- Check the ratios for geometric:
- [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- [tex]\( \frac{14}{7} = 2 \)[/tex]
- The ratios are equal, so this sequence is Geometric.
4. Sequence 4: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Check arithmetic differences:
- [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
- [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
- [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
- The differences are consistent. Hence, this sequence is Arithmetic.
- No need to check for geometric since it fits the arithmetic pattern.
5. Sequence 5: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Check arithmetic differences:
- [tex]\( 1 - (-1) = 2 \)[/tex]
- [tex]\( -1 - 1 = -2 \)[/tex]
- [tex]\( 1 - (-1) = 2 \)[/tex]
- The differences alternate.
- Check geometric ratios:
- [tex]\( \frac{1}{-1} = -1 \)[/tex]
- [tex]\( \frac{-1}{1} = -1 \)[/tex]
- The ratios alternate in sign.
- This alternating pattern does not fit a standard arithmetic or geometric sequence. Hence, it is Neither.
Now, summarizing the results:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Neither
1. Sequence 1: [tex]\( 98.3, 94.1, 89.9, 85.7, \ldots \)[/tex]
- To check if a sequence is arithmetic, we find the common difference by subtracting the first term from the second, the second term from the third, and so on.
- [tex]\( 94.1 - 98.3 = -4.2 \)[/tex]
- [tex]\( 89.9 - 94.1 = -4.2 \)[/tex]
- [tex]\( 85.7 - 89.9 = -4.2 \)[/tex]
- Each difference is the same, [tex]\(-4.2\)[/tex]. Hence, this sequence is Arithmetic.
2. Sequence 2: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- Let's check if it's arithmetic or geometric.
- For arithmetic, we calculate differences:
- [tex]\( 0 - 1 = -1 \)[/tex]
- [tex]\( -1 - 0 = -1 \)[/tex] (this seems incorrect as -1 - 0 = -1)
- [tex]\( 0 - (-1) = 1 \)[/tex]
- The differences are not consistent, so it is not an arithmetic sequence.
- Now for a geometric sequence, we check the ratios:
- [tex]\( \frac{0}{1} = 0 \)[/tex] (undefined ratio due to division by zero)
- The ratios aren't consistent either.
- This sequence is alternating, not fitting arithmetic or geometric patterns. Hence, it is Neither.
3. Sequence 3: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Check the differences for arithmetic:
- [tex]\( 3.5 - 1.75 = 1.75 \)[/tex]
- [tex]\( 7 - 3.5 = 3.5 \)[/tex]
- [tex]\( 14 - 7 = 7 \)[/tex]
- The differences aren't consistent, so it is not arithmetic.
- Check the ratios for geometric:
- [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- [tex]\( \frac{14}{7} = 2 \)[/tex]
- The ratios are equal, so this sequence is Geometric.
4. Sequence 4: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Check arithmetic differences:
- [tex]\( -10.8 - (-12) = 1.2 \)[/tex]
- [tex]\( -9.6 - (-10.8) = 1.2 \)[/tex]
- [tex]\( -8.4 - (-9.6) = 1.2 \)[/tex]
- The differences are consistent. Hence, this sequence is Arithmetic.
- No need to check for geometric since it fits the arithmetic pattern.
5. Sequence 5: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Check arithmetic differences:
- [tex]\( 1 - (-1) = 2 \)[/tex]
- [tex]\( -1 - 1 = -2 \)[/tex]
- [tex]\( 1 - (-1) = 2 \)[/tex]
- The differences alternate.
- Check geometric ratios:
- [tex]\( \frac{1}{-1} = -1 \)[/tex]
- [tex]\( \frac{-1}{1} = -1 \)[/tex]
- The ratios alternate in sign.
- This alternating pattern does not fit a standard arithmetic or geometric sequence. Hence, it is Neither.
Now, summarizing the results:
- Sequence 1: Arithmetic
- Sequence 2: Neither
- Sequence 3: Geometric
- Sequence 4: Arithmetic
- Sequence 5: Neither
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