type the correct answer in the box. use numerals instead of words.

amanda teaches the art of quilling to 4 students. these students each teach the art of quilling to 4 other students. this process continues through 3 more generations.

use a geometric series to model the number of people who know the art of quilling.

at the end of the process total people know the art of quilling.

Question

type the correct answer in the box. use numerals instead of words.

amanda teaches the art of quilling to 4 students. these students each teach the art of quilling to 4 other students. this process continues through 3 more generations.

use a geometric series to model the number of people who know the art of quilling.

at the end of the process   total people know the art of quilling.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Geometric Series", "Sum of Geometric Series" ], "current_concepts": [ "Geometric Series", "Sum of Geometric Series" ] } ## Identify the terms of the geometric series $$ \begin{aligned} &\text{Generation 0 (Amanda): } a_0 = 1 \ &\text{Generation 1 (Amanda's students): } a_1 = 4 \ &\text{Generation 2: } a_2 = 4 \times 4 = 16 \ &\text{Total generations: } 1 \text{ (initial) } + 3 \text{ more} = 4 \text{ student generations (Generations 1 to 4)} \end{aligned} $$ ## Calculate the sum of the geometric series $$ \begin{aligned} &S_n = \sum_{i=0}^{4} 4^i = 1 + 4 + 16 + 64 + 256 \ &S_n = \frac{1(4^5 - 1)}{4 - 1} = \frac{1024 - 1}{3} = \frac{1023}{3} = 341 \end{aligned} $$ At the end of the process 341 total people know the art of quilling. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Geometric Series" ] }

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Explanation:

Identify the terms of the geometric series

Calculate the sum of the geometric series

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