state whether the sequence, whose $n$th terms are indicated, is bounded and whether it is eventually monotone, increasing, or decreasing. (select all that apply)

\\frac{n}{7^n}, n \ge 2\

bounded
not bounded
monotone
not monotone
increasing
decreasing

Question

state whether the sequence, whose $n$th terms are indicated, is bounded and whether it is eventually monotone, increasing, or decreasing. (select all that apply)

\\frac{n}{7^n}, n \ge 2\

bounded
  not bounded
  monotone
  not monotone
  increasing
  decreasing

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Sequence Terms" ], "new_concepts": [ "Bounded Sequences", "Monotone Sequences" ], "current_concepts": [ "Sequence Terms", "Bounded Sequences", "Monotone Sequences" ] } ## Analyze monotonicity $$ \begin{aligned} a_n &= \frac{n}{7^n}, \quad n \ge 2 \ \frac{a_{n+1}}{a_n} &= \frac{n+1}{7^{n+1}} \cdot \frac{7^n}{n} = \frac{n+1}{7n} = \frac{1}{7} + \frac{1}{7n} \ \text{For } n \ge 2, \quad \frac{a_{n+1}}{a_n} &\le \frac{1}{7} + \frac{1}{14} = \frac{3}{14} < 1 \ a_{n+1} &< a_n \quad \forall n \ge 2 \end{aligned} $$ ## Determine boundedness $$ \begin{aligned} a_n &> 0 \quad \forall n \ge 2 \ a_2 &= \frac{2}{49} \ 0 < a_n &\le \frac{2}{49} \quad \forall n \ge 2 \end{aligned} $$ ## Select matching properties $$ \begin{aligned} &\text{Since } a_{n+1} < a_n, \text{ the sequence is decreasing and monotone.} \ &\text{Since } 0 < a_n \le \frac{2}{49}, \text{ the sequence is bounded.} \end{aligned} $$

bounded

not bounded

monotone

not monotone

increasing

decreasing

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "Monotone and Bounded Sequences" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze monotonicity

Determine boundedness

Select matching properties

Answer: