Skip to main content
Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.1.57a
Chapter 10, Problem 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

Verified step by step guidance
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:
\[h_0 = 20 \times 0.5^0\]
\[h_1 = 20 \times 0.5^1\]
\[h_2 = 20 \times 0.5^2\]
\[h_3 = 20 \times 0.5^3\]

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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 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.
Recommended video:
Guided course
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.
Recommended video:
Guided course
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.
Recommended video:
Guided course
7:39
Introduction to Exponent Rules
Related Practice
Textbook Question

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.

1
views
Textbook Question

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.

1
views
Textbook Question

67–70. Formulas for sequences of partial sums Consider the following infinite series.


a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Textbook Question

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

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

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.