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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 10.1.33a
Chapter 10, Problem 10.1.33a

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
a. Find the next two terms of the sequence.
{-5, 5, -5, 5, ......}

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Observe the given sequence: \(\{-5, 5, -5, 5, \ldots\}\). Notice the pattern of signs and values alternating between \(-5\) and \(5\).
Identify the rule governing the sequence. Since the terms alternate between \(-5\) and \(5\), the sequence can be described as \(a_n = (-1)^n \times 5\) or \(a_n = (-1)^{n+1} \times 5\), depending on the starting index.
To find the next two terms, determine the position of the last given term. The last term provided is the 4th term, which is \(5\).
Calculate the 5th term by applying the pattern: since the 4th term is \(5\), the 5th term will be \(-5\) (continuing the alternating sign pattern).
Similarly, calculate the 6th term, which will be \(5\), following the established alternating pattern.

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Key Concepts

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

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 its position. Understanding how terms relate to each other helps predict future terms.
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Pattern Recognition in Sequences

Identifying the pattern or rule governing the sequence is essential. This involves observing how terms change from one to the next, such as alternating signs or repeating values, to determine the next terms accurately.
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Alternating Sequences

An alternating sequence is one where the signs of the terms switch between positive and negative in a regular pattern. Recognizing this helps in predicting subsequent terms by applying the sign change rule consistently.
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Related Practice
Textbook Question

Explain why or why not

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

a.If limₙ→∞aₙ = 1 and limₙ→∞bₙ = 3, then limₙ→∞(bₙ / aₙ) = 3.

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Textbook Question

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


a. Find an upper bound for the remainder in terms of n.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

Textbook Question

{Use of Tech} Periodic dosing

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.


a.Find a recurrence relation for the sequence {dₙ} that gives the amount of drug in the blood after the nᵗʰ dose, where d₁ = 80.

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Textbook Question

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


a. Find an upper bound for the remainder in terms of n.


41. ∑ (k = 1 to ∞) 1 / k⁶

Textbook Question

Loglog p-series Consider the series ∑ (k = 2 to ∞) 1 / (k(ln k)(ln ln k)ᵖ), where p is a real number.

a. For what values of p does this series converge?

Textbook Question

{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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