Answer :
Certainly! Let's classify each of the sequences as arithmetic, geometric, or neither. Here’s how we do it:
1. Sequence 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 consistent difference between consecutive terms.
- Subtract the second term from the first term: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex].
- Subtract the third term from the second term: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex].
- Subtract the fourth term from the third term: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex].
- Since the difference is constant (-4.2), this sequence is arithmetic.
2. Sequence 2: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- Look at the differences between consecutive terms:
- [tex]\( 0 - 1 = -1 \)[/tex]
- [tex]\( -1 - 0 = -1 \)[/tex]
- The pattern doesn't hold consistently in an arithmetic sense.
- Check if there's a consistent ratio (divide each term by the previous non-zero term):
- Starting next to the first: [tex]\( \frac{0}{1} = 0 \)[/tex]
- Next, [tex]\( \frac{-1}{0} \)[/tex] is undefined.
- However, as we see the pattern of repeating [tex]\( 1, 0, -1, 0 \)[/tex], these terms repeat and don’t form a consistent ratio or difference that holds using standard definitions. Thus, the sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Check the ratio between successive terms:
- [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- [tex]\( \frac{14}{7} = 2 \)[/tex]
- Since there is a consistent ratio of 2, this sequence is geometric.
4. Sequence 4: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Check the differences between consecutive terms:
- [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 difference is consistent, this sequence is arithmetic.
5. Sequence 5: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Look at the differences between consecutive terms. Alternating between values doesn’t give a consistent difference in an arithmetic sense.
- Check if there's a consistent ratio:
- Since alternating signs and repeating [tex]\( -1 \)[/tex] and [tex]\( 1 \)[/tex] exist, we consider recurring patterns.
- The sequence is neither forming consistent differences nor consistent ratios relevant to standard definitions.
- The recurring flip in sign suggests a non-conventional sequence, so technically, it would often be treated as neither. However, inherent flips can demonstrate pattern existence, but wouldn't fit arithmetic or standard geometric.
In summary, based on our analysis, here's the classification:
- Sequence 1 is Arithmetic.
- Sequence 2 is Neither (due to its repeating pattern).
- Sequence 3 is Geometric.
- Sequence 4 is Arithmetic.
- Sequence 5 is Neither (due to alternate sign change repeats, often geometric interpretations with specific scenarios).
Let me know if you have more questions!
1. Sequence 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 consistent difference between consecutive terms.
- Subtract the second term from the first term: [tex]\( 94.1 - 98.3 = -4.2 \)[/tex].
- Subtract the third term from the second term: [tex]\( 89.9 - 94.1 = -4.2 \)[/tex].
- Subtract the fourth term from the third term: [tex]\( 85.7 - 89.9 = -4.2 \)[/tex].
- Since the difference is constant (-4.2), this sequence is arithmetic.
2. Sequence 2: [tex]\( 1, 0, -1, 0, \ldots \)[/tex]
- Look at the differences between consecutive terms:
- [tex]\( 0 - 1 = -1 \)[/tex]
- [tex]\( -1 - 0 = -1 \)[/tex]
- The pattern doesn't hold consistently in an arithmetic sense.
- Check if there's a consistent ratio (divide each term by the previous non-zero term):
- Starting next to the first: [tex]\( \frac{0}{1} = 0 \)[/tex]
- Next, [tex]\( \frac{-1}{0} \)[/tex] is undefined.
- However, as we see the pattern of repeating [tex]\( 1, 0, -1, 0 \)[/tex], these terms repeat and don’t form a consistent ratio or difference that holds using standard definitions. Thus, the sequence is neither arithmetic nor geometric.
3. Sequence 3: [tex]\( 1.75, 3.5, 7, 14 \)[/tex]
- Check the ratio between successive terms:
- [tex]\( \frac{3.5}{1.75} = 2 \)[/tex]
- [tex]\( \frac{7}{3.5} = 2 \)[/tex]
- [tex]\( \frac{14}{7} = 2 \)[/tex]
- Since there is a consistent ratio of 2, this sequence is geometric.
4. Sequence 4: [tex]\( -12, -10.8, -9.6, -8.4 \)[/tex]
- Check the differences between consecutive terms:
- [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 difference is consistent, this sequence is arithmetic.
5. Sequence 5: [tex]\( -1, 1, -1, 1, \ldots \)[/tex]
- Look at the differences between consecutive terms. Alternating between values doesn’t give a consistent difference in an arithmetic sense.
- Check if there's a consistent ratio:
- Since alternating signs and repeating [tex]\( -1 \)[/tex] and [tex]\( 1 \)[/tex] exist, we consider recurring patterns.
- The sequence is neither forming consistent differences nor consistent ratios relevant to standard definitions.
- The recurring flip in sign suggests a non-conventional sequence, so technically, it would often be treated as neither. However, inherent flips can demonstrate pattern existence, but wouldn't fit arithmetic or standard geometric.
In summary, based on our analysis, here's the classification:
- Sequence 1 is Arithmetic.
- Sequence 2 is Neither (due to its repeating pattern).
- Sequence 3 is Geometric.
- Sequence 4 is Arithmetic.
- Sequence 5 is Neither (due to alternate sign change repeats, often geometric interpretations with specific scenarios).
Let me know if you have more questions!
