select the sequences that are geometric.

18, 36, 54, 72, ...
4.1, 8.2, 16.4, 32.8, ...
-7, 14, -28, 56, ...
980, 784, 627.2, 501.76, ...
5, 2, -1, -4, ...

Question

select the sequences that are geometric.

18, 36, 54, 72, ...
  4.1, 8.2, 16.4, 32.8, ...
  -7, 14, -28, 56, ...
  980, 784, 627.2, 501.76, ...
  5, 2, -1, -4, ...

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Geometric Sequences", "Common Ratio" ], "new_concepts": [], "current_concepts": [ "Geometric Sequences", "Common Ratio", "Arithmetic Sequences" ] } ## Define geometric sequence condition Using the *Geometric Sequences* and *Common Ratio* knowledge points $$ r = \frac{a_{n}}{a_{n-1}} \quad \text{for all } n \ge 2 $$ ## Analyze the first sequence Using the *Geometric Sequences* and *Common Ratio* knowledge points $$ \begin{aligned} &\text{Sequence: } 18, 36, 54, 72, \dots \ &\frac{36}{18} = 2, \quad \frac{54}{36} = 1.5 \ &2 e 1.5 \quad (\text{Not geometric}) \end{aligned} $$ ## Analyze the second sequence Using the *Geometric Sequences* and *Common Ratio* knowledge points $$ \begin{aligned} &\text{Sequence: } 4.1, 8.2, 16.4, 32.8, \dots \ &\frac{8.2}{4.1} = 2, \quad \frac{16.4}{8.2} = 2, \quad \frac{32.8}{16.4} = 2 \ &r = 2 \quad (\text{Geometric}) \end{aligned} $$ ## Analyze the third sequence Using the *Geometric Sequences* and *Common Ratio* knowledge points $$ \begin{aligned} &\text{Sequence: } -7, 14, -28, 56, \dots \ &\frac{14}{-7} = -2, \quad \frac{-28}{14} = -2, \quad \frac{56}{-28} = -2 \ &r = -2 \quad (\text{Geometric}) \end{aligned} $$ ## Analyze the fourth and fifth sequences Using the *Geometric Sequences* and *Common Ratio* knowledge points $$ \begin{aligned} &\text{Sequence 4: } 980, 784, 627.2, 501.76, \dots \ &\frac{784}{980} = 0.8, \quad \frac{627.2}{784} = 0.8, \quad \frac{501.76}{627.2} = 0.8 \ &r = 0.8 \quad (\text{Geometric}) \ &\text{Sequence 5: } 5, 2, -1, -4, \dots \ &\frac{2}{5} = 0.4, \quad \frac{-1}{2} = -0.5 \ &0.4 e -0.5 \quad (\text{Not geometric}) \end{aligned} $$

18, 36, 54, 72, ...

4.1, 8.2, 16.4, 32.8, ...

-7, 14, -28, 56, ...

980, 784, 627.2, 501.76, ...

5, 2, -1, -4, ...

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Geometric Sequences" ] }

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Question

Explanation:

Define geometric sequence condition

Analyze the first sequence

Analyze the second sequence

Analyze the third sequence

Analyze the fourth and fifth sequences

Answer: