To determine if each given sequence is a geometric sequence, we need to analyze the common ratio between consecutive terms in each sequence.
A geometric sequence is one in which the ratio between successive terms is constant. This ratio is called the common ratio `r`.
Let's analyze each sequence:
Sequence A: 1, 3, 9, 27, 81
- Calculate the ratio between the second term and the first term: [tex]\( \frac{3}{1} = 3 \)[/tex]
- Check if this ratio is maintained throughout the sequence:
- [tex]\( \frac{9}{3} = 3 \)[/tex]
- [tex]\( \frac{27}{9} = 3 \)[/tex]
- [tex]\( \frac{81}{27} = 3 \)[/tex]
Since the ratio is constant, sequence A is a geometric sequence.
Sequence B: 10, 5, 2.5, 1.25, 0.625, 0.3125
- Calculate the ratio between the second term and the first term: [tex]\( \frac{5}{10} = 0.5 \)[/tex]
- Check if this ratio is maintained throughout the sequence:
- [tex]\( \frac{2.5}{5} = 0.5 \)[/tex]
- [tex]\( \frac{1.25}{2.5} = 0.5 \)[/tex]
- [tex]\( \frac{0.625}{1.25} = 0.5 \)[/tex]
- [tex]\( \frac{0.3125}{0.625} = 0.5 \)[/tex]
Since the ratio is constant, sequence B is a geometric sequence.
Sequence C: 5, 10, 20, 40, 80, 160
- Calculate the ratio between the second term and the first term: [tex]\( \frac{10}{5} = 2 \)[/tex]
- Check if this ratio is maintained throughout the sequence:
- [tex]\( \frac{20}{10} = 2 \)[/tex]
- [tex]\( \frac{40}{20} = 2 \)[/tex]
- [tex]\( \frac{80}{40} = 2 \)[/tex]
- [tex]\( \frac{160}{80} = 2 \)[/tex]
Since the ratio is constant, sequence C is a geometric sequence.
Sequence D: 3, 6, 9, 12, 15, 18
- Calculate the ratio between the second term and the first term: [tex]\( \frac{6}{3} = 2 \)[/tex]
- Check if this ratio is maintained throughout the sequence:
- [tex]\( \frac{9}{6} = 1.5 \)[/tex]
- [tex]\( \frac{12}{9} = 1.\overline{3} \)[/tex]
- [tex]\( \frac{15}{12} = 1.25 \)[/tex]
- [tex]\( \frac{18}{15} = 1.2 \)[/tex]
Since the ratio is not constant, sequence D is not a geometric sequence.
Conclusion:
- Sequence A is a geometric sequence.
- Sequence B is a geometric sequence.
- Sequence C is a geometric sequence.
- Sequence D is not a geometric sequence.
Thus, the geometric sequences are A, B, and C.
