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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 10.1.31a
Capitolo 10, Problema 10.1.31a

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, ......}

Guida verificata passo dopo passo
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Identify the pattern in the given sequence: {1, 3, 9, 27, 81, ......}. Notice how each term relates to the previous term.
Check if the sequence is geometric by dividing each term by the previous term. For example, calculate \( \frac{3}{1} \), \( \frac{9}{3} \), \( \frac{27}{9} \), and \( \frac{81}{27} \).
If the ratio between consecutive terms is constant, denote this common ratio as \( r \). This means the sequence is geometric and each term can be expressed as \( a_n = a_1 \times r^{n-1} \).
Use the common ratio \( r \) to find the next two terms by multiplying the last known term by \( r \) to get the next term, and then multiply that result by \( r \) again to get the term after that.
Write the expressions for the next two terms as \( a_6 = a_5 \times r \) and \( a_7 = a_6 \times r \), substituting the known values to express these terms explicitly.

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Sequences and Terms

A sequence is an ordered list of numbers following a specific pattern. Each number in the sequence is called a term, typically denoted as aₙ, where n indicates the term's position. Understanding how terms relate helps predict future terms.
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Introduction to Sequences

Geometric Sequences

A geometric sequence is one where each term is found by multiplying the previous term by a constant ratio. Identifying this ratio allows you to generate subsequent terms by repeated multiplication.
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Geometric Sequences - Recursive Formula

Pattern Recognition and Prediction

Recognizing the pattern in a sequence is essential to find missing or future terms. This involves analyzing the given terms, determining the rule (such as addition or multiplication), and applying it to extend the sequence.
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Sigma Notation Example 1
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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

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


a. The sum ∑ (k = 1 to ∞) 1 / 3ᵏ is a p-series.

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87. Explain why or why not

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


a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

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{Use of Tech} Drug Dosing

A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.



a.Let dₙ equal the amount of medication (in mg) in the bloodstream after n doses, where d₁ = 75.

Find a recurrence relation for dₙ.

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


a. n!n! = (2n)! for all positive integers n.

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{Use of Tech} Repeated square roots

Consider the sequence defined by

aₙ₊₁ = √(2 + aₙ),a₀ = √2, for n = 0, 1, 2, 3, …


a.Evaluate the first four terms of the sequence {aₙ}.

State the exact values first, and then the approximate values.

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