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Let's solve each sequence step-by-step:
1. Sequence [tex]\(\{23, 28, 33, 38\}\)[/tex]:
This sequence has a constant difference between consecutive terms. We can calculate the common difference ([tex]\(d\)[/tex]) as follows:
[tex]\[ d = 28 - 23 = 5 \][/tex]
Hence, the common difference for this arithmetic sequence is [tex]\(5\)[/tex].
2. Sequence [tex]\(\left\{\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{8}{9}\right\}\)[/tex]:
This sequence consists of fractions with increasing numerators and denominators. Each term is:
[tex]\[ \frac{n+1}{n+2} \quad \text{where } n = 2, 3, 4, \ldots 8 \][/tex]
These are all terms in simplified fractional form and the sequence continues in this progression.
3. Sequence [tex]\(\{34, 47, 62\}\)[/tex]:
This sequence also has a constant difference. We can calculate the common difference ([tex]\(d\)[/tex]) as follows:
[tex]\[ d = 47 - 34 = 13 \][/tex]
Hence, the common difference for this arithmetic sequence is [tex]\(13\)[/tex].
4. Sequence [tex]\(\left\{\frac{2}{5}, \frac{1}{6}, \square, -\frac{1}{7}, \square, \square\right\}\)[/tex]:
To find the missing terms, we assume it is an arithmetic progression and calculate the initial difference between the first two known terms:
[tex]\[ \text{Difference} = \frac{1}{6} - \frac{2}{5} = \frac{5 - 12}{30} = -\frac{7}{30} \][/tex]
Using this difference, we find the subsequent terms by sequentially subtracting [tex]\( -\frac{7}{30} \)[/tex]:
[tex]\[ \text{Third term} = \frac{1}{6} - \frac{7}{30} = \frac{5 - 7}{30} = -\frac{2}{30} = -\frac{1}{15} \][/tex]
Next, determine the fifth and sixth terms:
[tex]\[ \text{Fifth term} = -\frac{1}{7} - \frac{7}{30} = -\frac{30+7}{210} = -\frac{37}{210} = -0.1761904761904762 \][/tex]
[tex]\[ \text{Sixth term} = -0.1761904761904762 - 0.23333333333333334 = -0.4095238095238096 \][/tex]
So, the sequence is now:
[tex]\(\left\{\frac{2}{5}, \frac{1}{6}, -\frac{1}{15}, -\frac{1}{7}, -\frac{37}{210}, -\frac{86}{210}\right\}\)[/tex]
5. Sequence [tex]\(\{16, 217\}\)[/tex]:
This sequence does not have enough information provided to determine if it follows any specific progression. Therefore, we list it as it is:
[tex]\(\{16, 217\}\)[/tex]
In summary:
- Sequence 1: [tex]\(\{23, 28, 33, 38\}\)[/tex], with a common difference of [tex]\(5\)[/tex].
- Sequence 2: [tex]\(\left\{\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{8}{9}\right\}\)[/tex].
- Sequence 3: [tex]\(\{34, 47, 62\}\)[/tex], with a common difference of [tex]\(13\)[/tex].
- Sequence 4: [tex]\(\left\{\frac{2}{5}, \frac{1}{6}, -\frac{1}{15}, -\frac{1}{7}, -\frac{37}{210}, -\frac{86}{210}\right\}\)[/tex].
- Sequence 5: [tex]\(\{16, 217\}\)[/tex].
1. Sequence [tex]\(\{23, 28, 33, 38\}\)[/tex]:
This sequence has a constant difference between consecutive terms. We can calculate the common difference ([tex]\(d\)[/tex]) as follows:
[tex]\[ d = 28 - 23 = 5 \][/tex]
Hence, the common difference for this arithmetic sequence is [tex]\(5\)[/tex].
2. Sequence [tex]\(\left\{\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{8}{9}\right\}\)[/tex]:
This sequence consists of fractions with increasing numerators and denominators. Each term is:
[tex]\[ \frac{n+1}{n+2} \quad \text{where } n = 2, 3, 4, \ldots 8 \][/tex]
These are all terms in simplified fractional form and the sequence continues in this progression.
3. Sequence [tex]\(\{34, 47, 62\}\)[/tex]:
This sequence also has a constant difference. We can calculate the common difference ([tex]\(d\)[/tex]) as follows:
[tex]\[ d = 47 - 34 = 13 \][/tex]
Hence, the common difference for this arithmetic sequence is [tex]\(13\)[/tex].
4. Sequence [tex]\(\left\{\frac{2}{5}, \frac{1}{6}, \square, -\frac{1}{7}, \square, \square\right\}\)[/tex]:
To find the missing terms, we assume it is an arithmetic progression and calculate the initial difference between the first two known terms:
[tex]\[ \text{Difference} = \frac{1}{6} - \frac{2}{5} = \frac{5 - 12}{30} = -\frac{7}{30} \][/tex]
Using this difference, we find the subsequent terms by sequentially subtracting [tex]\( -\frac{7}{30} \)[/tex]:
[tex]\[ \text{Third term} = \frac{1}{6} - \frac{7}{30} = \frac{5 - 7}{30} = -\frac{2}{30} = -\frac{1}{15} \][/tex]
Next, determine the fifth and sixth terms:
[tex]\[ \text{Fifth term} = -\frac{1}{7} - \frac{7}{30} = -\frac{30+7}{210} = -\frac{37}{210} = -0.1761904761904762 \][/tex]
[tex]\[ \text{Sixth term} = -0.1761904761904762 - 0.23333333333333334 = -0.4095238095238096 \][/tex]
So, the sequence is now:
[tex]\(\left\{\frac{2}{5}, \frac{1}{6}, -\frac{1}{15}, -\frac{1}{7}, -\frac{37}{210}, -\frac{86}{210}\right\}\)[/tex]
5. Sequence [tex]\(\{16, 217\}\)[/tex]:
This sequence does not have enough information provided to determine if it follows any specific progression. Therefore, we list it as it is:
[tex]\(\{16, 217\}\)[/tex]
In summary:
- Sequence 1: [tex]\(\{23, 28, 33, 38\}\)[/tex], with a common difference of [tex]\(5\)[/tex].
- Sequence 2: [tex]\(\left\{\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{8}{9}\right\}\)[/tex].
- Sequence 3: [tex]\(\{34, 47, 62\}\)[/tex], with a common difference of [tex]\(13\)[/tex].
- Sequence 4: [tex]\(\left\{\frac{2}{5}, \frac{1}{6}, -\frac{1}{15}, -\frac{1}{7}, -\frac{37}{210}, -\frac{86}{210}\right\}\)[/tex].
- Sequence 5: [tex]\(\{16, 217\}\)[/tex].
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