72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence

d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.

Drug elimination
Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

검증된 단계별 안내

Identify the type of sequence described. Since 5% of the drug is eliminated every hour, 95% remains each hour. This means the amount of drug decreases by a constant ratio, indicating a geometric sequence.

Write the general formula for the amount of drug after n hours. Since the initial amount is \(d_0 = 200\) mg and the drug decreases by 5% each hour, the formula is: \(d_n = 200 \times (0.95)^n\)

Understand the behavior of the sequence as \(n\) becomes very large. Since \(0 < 0.95 < 1\), the term \((0.95)^n\) gets smaller and smaller as \(n\) increases.

Use a calculator or graphing utility to evaluate \(d_n\) for large values of \(n\) to observe the trend. This will help estimate the limit of the sequence.

Conclude the limit based on the observed behavior: if \(d_n\) approaches a specific value as \(n\) approaches infinity, that value is the limit; otherwise, state that the limit does not exist.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant ratio. In this problem, the amount of drug decreases by a fixed percentage each hour, forming a geometric sequence with a common ratio less than 1.

Limit of a Sequence

The limit of a sequence describes the value that the terms approach as the index goes to infinity. For a geometric sequence with a ratio between -1 and 1, the limit exists and can be used to understand the long-term behavior of the drug concentration.

Use of Technology in Calculus

Calculators or graphing utilities can approximate sequence values and limits when analytical solutions are complex. They help visualize the sequence behavior over time and estimate limits, especially useful for real-world applications like drug elimination.