Answer :
To determine which of the given sequences is an arithmetic sequence, we need to verify whether the differences between consecutive terms in each sequence are consistent.
### Sequence 1: [tex]\(-\frac{7}{11}, \frac{6}{11},-\frac{5}{11}, \frac{4}{11}, \ldots\)[/tex]
Calculate the differences between consecutive terms:
1. [tex]\(\frac{6}{11} - \left(-\frac{7}{11}\right) = \frac{6}{11} + \frac{7}{11} = \frac{13}{11}\)[/tex]
2. [tex]\(-\frac{5}{11} - \frac{6}{11} = -\frac{5}{11} - \frac{6}{11} = -\frac{11}{11} = -1\)[/tex]
3. [tex]\(\frac{4}{11} - \left(-\frac{5}{11}\right) = \frac{4}{11} + \frac{5}{11} = \frac{9}{11}\)[/tex]
The differences are [tex]\(\frac{13}{11}, -1, \frac{9}{11}\)[/tex], which are not equal. Hence, this is not an arithmetic sequence.
### Sequence 2: [tex]\(-\frac{3}{4}, -\frac{3}{5}, -\frac{3}{6}, -\frac{3}{7}, \ldots\)[/tex]
Calculate the differences between consecutive terms:
1. [tex]\(-\frac{3}{5} - \left(-\frac{3}{4}\right) = -\frac{3}{5} + \frac{3}{4} = \frac{3}{4} - \frac{3}{5}\)[/tex]
[tex]\[ = \frac{15}{20} - \frac{12}{20} = \frac{3}{20} \][/tex]
2. [tex]\(-\frac{3}{6} - \left(-\frac{3}{5}\right) = -\frac{3}{6} + \frac{3}{5} = \frac{3}{5} - \frac{3}{6}\)[/tex]
[tex]\[ = \frac{18}{30} - \frac{15}{30} = \frac{3}{30} = \frac{1}{10} \][/tex]
3. [tex]\(-\frac{3}{7} - \left(-\frac{3}{6}\right) = -\frac{3}{7} + \frac{3}{6} = \frac{3}{6} - \frac{3}{7}\)[/tex]
[tex]\[ = \frac{21}{42} - \frac{18}{42} = \frac{3}{42} = \frac{1}{14} \][/tex]
The differences are [tex]\(\frac{3}{20}, \frac{1}{10}, \frac{1}{14}\)[/tex], which are not equal. Hence, this is not an arithmetic sequence.
### Sequence 3: [tex]\(\frac{1}{2}, 2, \frac{7}{2}, 5, \ldots\)[/tex]
Calculate the differences between consecutive terms:
1. [tex]\(2 - \frac{1}{2} = 2 - 0.5 = 1.5 = \frac{3}{2}\)[/tex]
2. [tex]\(\frac{7}{2} - 2 = \frac{7}{2} - \frac{4}{2} = \frac{3}{2}\)[/tex]
3. [tex]\(5 - \frac{7}{2} = 5 - 3.5 = 1.5 = \frac{3}{2}\)[/tex]
The differences are [tex]\(\frac{3}{2}, \frac{3}{2}, \frac{3}{2}\)[/tex], which are all equal. Hence, this is an arithmetic sequence.
### Sequence 4: [tex]\(\frac{3}{4}, -\frac{3}{2}, 3, -6, \ldots\)[/tex]
Calculate the differences between consecutive terms:
1. [tex]\(-\frac{3}{2} - \frac{3}{4} = -\frac{3}{2} - \frac{3}{4} = -\frac{6}{4} - \frac{3}{4} = -\frac{9}{4}\)[/tex]
2. [tex]\(3 - \left(-\frac{3}{2}\right) = 3 + \frac{3}{2} = \frac{6}{2} + \frac{3}{2} = \frac{9}{2}\)[/tex]
3. [tex]\(-6 - 3 = -9\)[/tex]
The differences are [tex]\(-\frac{9}{4}, \frac{9}{2}, -9\)[/tex], which are not equal. Hence, this is not an arithmetic sequence.
### Conclusion
Among the given sequences, only the third sequence [tex]\(\frac{1}{2}, 2, \frac{7}{2}, 5, \ldots\)[/tex] is an arithmetic sequence.
### Sequence 1: [tex]\(-\frac{7}{11}, \frac{6}{11},-\frac{5}{11}, \frac{4}{11}, \ldots\)[/tex]
Calculate the differences between consecutive terms:
1. [tex]\(\frac{6}{11} - \left(-\frac{7}{11}\right) = \frac{6}{11} + \frac{7}{11} = \frac{13}{11}\)[/tex]
2. [tex]\(-\frac{5}{11} - \frac{6}{11} = -\frac{5}{11} - \frac{6}{11} = -\frac{11}{11} = -1\)[/tex]
3. [tex]\(\frac{4}{11} - \left(-\frac{5}{11}\right) = \frac{4}{11} + \frac{5}{11} = \frac{9}{11}\)[/tex]
The differences are [tex]\(\frac{13}{11}, -1, \frac{9}{11}\)[/tex], which are not equal. Hence, this is not an arithmetic sequence.
### Sequence 2: [tex]\(-\frac{3}{4}, -\frac{3}{5}, -\frac{3}{6}, -\frac{3}{7}, \ldots\)[/tex]
Calculate the differences between consecutive terms:
1. [tex]\(-\frac{3}{5} - \left(-\frac{3}{4}\right) = -\frac{3}{5} + \frac{3}{4} = \frac{3}{4} - \frac{3}{5}\)[/tex]
[tex]\[ = \frac{15}{20} - \frac{12}{20} = \frac{3}{20} \][/tex]
2. [tex]\(-\frac{3}{6} - \left(-\frac{3}{5}\right) = -\frac{3}{6} + \frac{3}{5} = \frac{3}{5} - \frac{3}{6}\)[/tex]
[tex]\[ = \frac{18}{30} - \frac{15}{30} = \frac{3}{30} = \frac{1}{10} \][/tex]
3. [tex]\(-\frac{3}{7} - \left(-\frac{3}{6}\right) = -\frac{3}{7} + \frac{3}{6} = \frac{3}{6} - \frac{3}{7}\)[/tex]
[tex]\[ = \frac{21}{42} - \frac{18}{42} = \frac{3}{42} = \frac{1}{14} \][/tex]
The differences are [tex]\(\frac{3}{20}, \frac{1}{10}, \frac{1}{14}\)[/tex], which are not equal. Hence, this is not an arithmetic sequence.
### Sequence 3: [tex]\(\frac{1}{2}, 2, \frac{7}{2}, 5, \ldots\)[/tex]
Calculate the differences between consecutive terms:
1. [tex]\(2 - \frac{1}{2} = 2 - 0.5 = 1.5 = \frac{3}{2}\)[/tex]
2. [tex]\(\frac{7}{2} - 2 = \frac{7}{2} - \frac{4}{2} = \frac{3}{2}\)[/tex]
3. [tex]\(5 - \frac{7}{2} = 5 - 3.5 = 1.5 = \frac{3}{2}\)[/tex]
The differences are [tex]\(\frac{3}{2}, \frac{3}{2}, \frac{3}{2}\)[/tex], which are all equal. Hence, this is an arithmetic sequence.
### Sequence 4: [tex]\(\frac{3}{4}, -\frac{3}{2}, 3, -6, \ldots\)[/tex]
Calculate the differences between consecutive terms:
1. [tex]\(-\frac{3}{2} - \frac{3}{4} = -\frac{3}{2} - \frac{3}{4} = -\frac{6}{4} - \frac{3}{4} = -\frac{9}{4}\)[/tex]
2. [tex]\(3 - \left(-\frac{3}{2}\right) = 3 + \frac{3}{2} = \frac{6}{2} + \frac{3}{2} = \frac{9}{2}\)[/tex]
3. [tex]\(-6 - 3 = -9\)[/tex]
The differences are [tex]\(-\frac{9}{4}, \frac{9}{2}, -9\)[/tex], which are not equal. Hence, this is not an arithmetic sequence.
### Conclusion
Among the given sequences, only the third sequence [tex]\(\frac{1}{2}, 2, \frac{7}{2}, 5, \ldots\)[/tex] is an arithmetic sequence.
