Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.1.59a

Chapter 10, Problem 10.1.59a

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 = 30,r = 0.25

Verified step by step guidance

Understand the problem: The ball is initially thrown to a height of \(h_0 = 30\) meters. After each bounce, it reaches a height that is a fraction \(r = 0.25\) 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 term \(h_0\): This is the initial height before any bounce, so \(h_0 = 30\) meters.

Calculate the second term \(h_1\): Use the formula \(h_1 = 30 \times 0.25^1\) to find the height after the first bounce.

Calculate the third and fourth terms \(h_2\) and \(h_3\): Similarly, use \(h_2 = 30 \times 0.25^2\) and \(h_3 = 30 \times 0.25^3\) to find the heights after the second and third bounces respectively.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first 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.

Initial Term and Common Ratio

The initial term (h₀) represents the starting height of the ball, while the common ratio (r) is the fraction of the height the ball reaches after each bounce. These two values define the entire sequence of heights.

Sequence Notation and Term Calculation

The nth term of a geometric sequence is given by hₙ = h₀ * rⁿ. Using this formula, you can calculate the height after any bounce by raising the ratio to the power of n and multiplying by the initial height.

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Textbook Question

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.

a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.

∑ (k = 0 to ∞) (–2/7)ᵏ

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Textbook Question

Explain why or why not

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

a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.

Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

a. Find the next two terms of the sequence.

{1, 3, 9, 27, 81, ......}

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Textbook Question

Find the first term a and the ratio r of each geometric series.

a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ

Textbook Question

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.

Textbook Question

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, …}.