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Question 7 of 10

Which of the following are geometric sequences? Check all that apply.

A. 1, 3, 9, 27, 81
B. 10, 5, 2.5, 1.25, 0.625, 0.3125
C. 5, 10, 20, 40, 80, 160
D. 3, 6, 9, 12, 15, 18

Sagot :

GinnyAnswer
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.
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