which of the following is a geometric sequence with a common ratio of 2?

a) 14, 28, 56, 112, ...
b) 87, 85, 83, 81, ...
c) 64, 32, 16, 8, ...
d) 14, 16, 18, 20, ...

question 8 (5 points)
evaluate the series: $\sum_{k=1}^{6} (25 - k^2)$

a) 129
b) 59
c) 84
d) 70

question 9 (5 points)
identify the common ratio of the sequence: 400, 200, 100, 50, ...

a) 4
b) $\frac{1}{4}$
c) $\frac{1}{2}$
d) 2

Question

which of the following is a geometric sequence with a common ratio of 2?

a) 14, 28, 56, 112, ...
b) 87, 85, 83, 81, ...
c) 64, 32, 16, 8, ...
d) 14, 16, 18, 20, ...

question 8 (5 points)
evaluate the series: $\sum_{k=1}^{6} (25 - k^2)$

a) 129
b) 59
c) 84
d) 70

question 9 (5 points)
identify the common ratio of the sequence: 400, 200, 100, 50, ...

a) 4
b) $\frac{1}{4}$
c) $\frac{1}{2}$
d) 2

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Sigma Notation" ], "new_concepts": [ "Geometric Sequences", "Common Ratio" ], "current_concepts": [ "Geometric Sequences", "Common Ratio", "Sigma Notation" ] } ## Identify geometric sequence with common ratio of 2 $$ \begin{aligned} &\text{Option A: } \frac{28}{14} = 2, \quad \frac{56}{28} = 2, \quad \frac{112}{56} = 2 \quad (\text{Geometric, } r = 2) \ &\text{Option B: } 85 - 87 = -2 \quad (\text{Arithmetic, } d = -2) \ &\text{Option C: } \frac{32}{64} = \frac{1}{2} \quad (\text{Geometric, } r = \frac{1}{2}) \ &\text{Option D: } 16 - 14 = 2 \quad (\text{Arithmetic, } d = 2) \end{aligned} $$ ## Evaluate the summation series $$ \begin{aligned} \sum_{k=1}^{6} (25 - k^2) &= (25 - 1^2) + (25 - 2^2) + (25 - 3^2) + (25 - 4^2) + (25 - 5^2) + (25 - 6^2) \ &= 24 + 21 + 16 + 9 + 0 + (-11) \ &= 59 \end{aligned} $$ ## Find the common ratio of the sequence $$ \begin{aligned} &\text{Sequence: } 400, 200, 100, 50, \dots \ &r = \frac{a_2}{a_1} = \frac{200}{400} = \frac{1}{2} \end{aligned} $$ ## Question 1 (A) 14, 28, 56, 112, ...

(B) 87, 85, 83, 81, ...

(C) 64, 32, 16, 8, ...

(D) 14, 16, 18, 20, ... ## Question 8 (A) 129

(B) 59

(C) 84

(D) 70 ## Question 9 (A) 4

(B) $\frac{1}{4}$

(C) $\frac{1}{2}$

(D) 2

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

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QUESTION IMAGE

Question

Explanation:

Identify geometric sequence with common ratio of 2

Evaluate the summation series

Find the common ratio of the sequence

Answer:

Question 1

Question 8

Question 9