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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 10.1.29c
Capitolo 10, Problema 10.1.29c

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
c. Find an explicit formula for the nth term of the sequence.


{1, 2, 4, 8, 16, ......}

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Identify the pattern in the given sequence: {1, 2, 4, 8, 16, ...}. Notice how each term relates to the previous one.
Observe that each term is obtained by multiplying the previous term by 2. This suggests the sequence is geometric with common ratio r = 2.
Recall the explicit formula for the nth term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio.
Substitute the known values into the formula: \(a_1 = 1\) and \(r = 2\), so the formula becomes \(a_n = 1 \times 2^{n-1}\).
Simplify the formula to get the explicit nth term: \(a_n = 2^{n-1}\). This formula allows you to find any term in the sequence directly.

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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 progress helps in identifying patterns and formulating explicit expressions.
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Introduction to Sequences

Explicit Formula for a Sequence

An explicit formula expresses the nth term of a sequence directly in terms of n, allowing calculation of any term without knowing previous terms. For example, geometric sequences have formulas like aₙ = a₁ * r^(n-1), where r is the common ratio.
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Arithmetic Sequences - General Formula

Geometric Sequences

A geometric sequence is one where each term is found by multiplying the previous term by a constant ratio. Recognizing this pattern is key to writing the explicit formula. In the given sequence {1, 2, 4, 8, 16, ...}, the ratio is 2, indicating a geometric progression.
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Geometric Sequences - Recursive Formula
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72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


c.Find a recurrence relation that generates 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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{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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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. A series that converges conditionally must converge.

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Explain why or why not

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



b.If limₙ→∞aₙ = 0 and limₙ→∞bₙ = ∞, then limₙ→∞aₙbₙ = 0.

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87. Explain why or why not

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


c. If ∑ aₖ converges, then ∑ (aₖ + 0.0001) converges.

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


c. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.

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