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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.1.31b
Capitolo 10, Problema 10.1.31b

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).


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

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1
Identify the pattern in the given sequence: {1, 3, 9, 27, 81, ...}. Notice how each term relates to the previous term.
Observe that each term is obtained by multiplying the previous term by 3. For example, 3 = 1 \(\times\) 3, 9 = 3 \(\times\) 3, and so on.
Express this relationship as a recurrence relation: \[a_{n} = 3 \times a_{n-1}\] where \[n \geq 2\].
Specify the initial condition, which is the first term of the sequence: \[a_1 = 1\].
Combine the recurrence relation and the initial condition to fully define the sequence: \[a_1 = 1, \quad a_n = 3a_{n-1} \text{ for } n \geq 2\].

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Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule. Each number in the sequence is called a term, denoted as aₙ, where n indicates the term's position. Understanding how terms relate to their positions is essential for analyzing and generating sequences.
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Introduction to Sequences

Recurrence Relation

A recurrence relation expresses each term of a sequence as a function of one or more previous terms. It provides a way to generate the sequence step-by-step, starting from initial term(s). Identifying the recurrence relation helps in understanding the pattern and behavior of the sequence.
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Percorso guidato
04:16
Intro To Related Rates

Initial Conditions

Initial conditions specify the starting term(s) of a sequence, which are necessary to uniquely determine all subsequent terms using the recurrence relation. Without these values, the sequence cannot be fully generated or analyzed.
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05:03
Initial Value Problems
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67–70. Formulas for sequences of partial sums Consider the following infinite series.


b.Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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Explain why or why not

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


b.If a sequence of positive numbers converges, then the sequence is decreasing.

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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.

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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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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.


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


b. Evaluate the series using Theorem 10.7.


∑ (k = 0 to ∞) (–2/7)ᵏ

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