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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 10.2.93b
Chapter 10, Problem 10.2.93b

Suppose the sequence {aₙ}⁽∞⁾ₙ₌₀ is defined by the recurrence relation
aₙ₊₁ = ⅓aₙ + 6;a₀ = 3.


b.Explain why {aₙ}⁽∞⁾ₙ₌₀ converges and find the limit.

Verified step by step guidance
1
Recognize that the sequence \( \{a_n\} \) is defined by a linear recurrence relation of the form \( a_{n+1} = r a_n + c \), where \( r = \frac{1}{3} \) and \( c = 6 \). Since \( |r| = \frac{1}{3} < 1 \), the sequence is a contraction and will converge to a fixed point.
To find the limit \( L \) of the sequence, assume it exists and satisfies the recurrence relation in the limit, so \( L = \frac{1}{3} L + 6 \). This is because as \( n \to \infty \), \( a_n \to L \) and \( a_{n+1} \to L \).
Solve the equation for \( L \): \( L = \frac{1}{3} L + 6 \). Rearranging gives \( L - \frac{1}{3} L = 6 \), which simplifies to \( \frac{2}{3} L = 6 \).
Multiply both sides by \( \frac{3}{2} \) to isolate \( L \), yielding \( L = 6 \times \frac{3}{2} \).
Interpret the result: since the sequence converges to \( L \), this value is the fixed point of the recurrence relation and represents the long-term behavior of \( \{a_n\} \).

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Key Concepts

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

Recurrence Relations

A recurrence relation defines each term of a sequence based on previous terms. Understanding how to manipulate and solve these relations is essential to analyze the behavior of sequences, such as finding explicit formulas or limits.
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Convergence of Sequences

A sequence converges if its terms approach a specific finite value as the index goes to infinity. Determining convergence involves analyzing the long-term behavior of the sequence, often by examining the recurrence relation or using limit properties.
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Introduction to Sequences

Finding Limits of Linear Recurrence Sequences

For linear recurrence relations like aₙ₊₁ = r aₙ + c with |r| < 1, the sequence converges to the fixed point L = c / (1 - r). This limit is found by setting aₙ₊₁ = aₙ = L and solving the resulting equation.
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Related Practice
Textbook Question

{Use of Tech} A savings plan

James begins a savings plan in which he deposits \(100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \)100.


Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \$0.


b.Find a recurrence relation that generates the sequence {Bₙ}.

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Textbook Question

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.


41. ∑ (k = 1 to ∞) 1 / k⁶

Textbook Question

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.


b. Find an explicit formula for the nth term of the sequence {hₙ}.


h₀ = 20,r = 0.5

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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.


b. Find an upper bound for the remainder Rₙ.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

Textbook Question

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.


b. Find an explicit formula for the nth term of the sequence {hₙ}.


h₀ = 30,r = 0.25

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Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


b.Find an explicit formula for the 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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