Answer :
To determine which of the given series is a geometric series, we need to examine the ratios between consecutive terms in each series. A series is geometric if the ratio between successive terms is constant.
Let's analyze each series one by one:
1. The series [tex]\(6, 13, 20, 27\)[/tex]:
[tex]\[ \frac{13}{6} \approx 2.167, \quad \frac{20}{13} \approx 1.538, \quad \frac{27}{20} = 1.35 \][/tex]
The ratios are not consistent, so this is not a geometric series.
2. The series [tex]\(7, 21, 35, 45\)[/tex]:
[tex]\[ \frac{21}{7} = 3, \quad \frac{35}{21} \approx 1.667, \quad \frac{45}{35} \approx 1.286 \][/tex]
The ratios are not consistent, so this is not a geometric series.
3. The series [tex]\(14, 21, 28, 35\)[/tex]:
[tex]\[ \frac{21}{14} = 1.5, \quad \frac{28}{21} \approx 1.333, \quad \frac{35}{28} = 1.25 \][/tex]
The ratios are not consistent, so this is not a geometric series.
4. The series [tex]\(2, 14, 98, 686\)[/tex]:
[tex]\[ \frac{14}{2} = 7, \quad \frac{98}{14} = 7, \quad \frac{686}{98} = 7 \][/tex]
The ratios are consistent and equal to 7, so this is a geometric series.
Hence, the series that forms a geometric progression is:
[tex]\[ 2, 14, 98, 686 \][/tex]
Let's analyze each series one by one:
1. The series [tex]\(6, 13, 20, 27\)[/tex]:
[tex]\[ \frac{13}{6} \approx 2.167, \quad \frac{20}{13} \approx 1.538, \quad \frac{27}{20} = 1.35 \][/tex]
The ratios are not consistent, so this is not a geometric series.
2. The series [tex]\(7, 21, 35, 45\)[/tex]:
[tex]\[ \frac{21}{7} = 3, \quad \frac{35}{21} \approx 1.667, \quad \frac{45}{35} \approx 1.286 \][/tex]
The ratios are not consistent, so this is not a geometric series.
3. The series [tex]\(14, 21, 28, 35\)[/tex]:
[tex]\[ \frac{21}{14} = 1.5, \quad \frac{28}{21} \approx 1.333, \quad \frac{35}{28} = 1.25 \][/tex]
The ratios are not consistent, so this is not a geometric series.
4. The series [tex]\(2, 14, 98, 686\)[/tex]:
[tex]\[ \frac{14}{2} = 7, \quad \frac{98}{14} = 7, \quad \frac{686}{98} = 7 \][/tex]
The ratios are consistent and equal to 7, so this is a geometric series.
Hence, the series that forms a geometric progression is:
[tex]\[ 2, 14, 98, 686 \][/tex]
