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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 10 - Sequences and Infinite Series문제 10.1.57a
10장, 문제 10.1.57a

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


a. Find the first four terms of the sequence of heights {hₙ}.


h₀ = 20,r = 0.5

검증된 단계별 안내
1
Understand the problem: The ball starts at an initial height \(h_0 = 20\) meters. After each bounce, it reaches a height that is a fraction \(r = 0.5\) of the previous height. We want to find the first four terms of the sequence \(\{h_n\}\), where \(h_n\) is the height after the \(n\)th bounce.
Recall the formula for the height after the \(n\)th bounce: \(h_n = h_0 \times r^n\). This means each term is the initial height multiplied by the rebound fraction raised to the power of the bounce number.
Calculate the first four terms by substituting \(n = 0, 1, 2, 3\) into the formula:
\[h_0 = 20 \times 0.5^0\]
\[h_1 = 20 \times 0.5^1\]
\[h_2 = 20 \times 0.5^2\]
\[h_3 = 20 \times 0.5^3\]

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m

주요 개념

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

Geometric Sequences

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant ratio. In this problem, the heights after each bounce form a geometric sequence with initial term h₀ and common ratio r, representing the fraction of height retained after each bounce.
추천 영상:
04:18
Geometric Sequences - Recursive Formula

Recursive and Explicit Formulas

Sequences can be described recursively, where each term depends on the previous one, or explicitly, where the nth term is given directly. Here, hₙ = r * hₙ₋₁ defines the recursive formula, and hₙ = h₀ * rⁿ is the explicit formula to find the height after the nth bounce.
추천 영상:
5:59
Recursive Formulas

Application of Exponents

Exponents are used to express repeated multiplication, essential for calculating terms in geometric sequences. Since each bounce height is multiplied by r repeatedly, exponents allow us to compute hₙ efficiently as h₀ times r raised to the power n.
추천 영상:
7:39
Introduction to Exponent Rules
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교과서 질문

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


a.Write out the first five terms of the sequence.


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.

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교과서 질문

71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.

a. Use a telescoping series argument.

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교과서 질문

39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.


a. Use Sₙ to estimate the sum of the series.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

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교과서 질문

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.

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교과서 질문

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

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교과서 질문

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


a.Write out the first five terms of the sequence.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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