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

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.1.73a

Capitolo 10, Problema 10.1.73a

72–75. {Use of Tech} Practical sequences
Consider the following situations that generate a sequence

a.Write out the first five 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.

Guida verificata passo dopo passo

Identify the type of sequence described. Since the material loses 50% of its mass every 10 years, this is a geometric sequence where each term is multiplied by a common ratio.

Determine the common ratio \( r \). Because the material retains 50% of its mass each decade, \( r = 0.5 \).

Write the general formula for the \( n^{th} \) term of the sequence: \( M_n = M_0 \times r^n \), where \( M_0 = 20 \) grams is the initial mass.

Calculate the first five terms by substituting \( n = 0, 1, 2, 3, 4 \) into the formula: \( M_0, M_1, M_2, M_3, M_4 \).

Express each term explicitly as \( M_n = 20 \times (0.5)^n \) grams, and write out the values for \( n = 0 \) through \( n = 4 \) without simplifying the numerical results.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant ratio. In this problem, the mass decreases by 50% every 10 years, so the ratio is 0.5. Understanding geometric sequences helps to write out terms like M₁ = M₀ × 0.5, M₂ = M₁ × 0.5, and so on.

Video consigliato:

04:18

Geometric Sequences - Recursive Formula

Exponential Decay

Exponential decay describes processes where quantities decrease at a rate proportional to their current value. Radioactive decay is a classic example, where the mass reduces by a fixed percentage over equal time intervals. This concept explains why the mass halves every decade, leading to a geometric sequence.

Video consigliato:

09:29

Exponential Growth & Decay

Sequence Notation and Indexing

Sequence notation uses subscripts to denote terms, such as Mₙ for the nth term. Indexing helps track the progression of the sequence over discrete steps—in this case, decades. Correctly interpreting M₀ as the initial mass and Mₙ as the mass after n decades is essential for writing and understanding the terms.

Video consigliato:

8:22

Introduction to Sequences

Pratica correlata

Domanda del libro di testo

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.

a. Find the first four terms of the sequence of heights {hₙ}.

h₀ = 30,r = 0.25

views

Domanda del libro di testo

Explain why or why not

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

a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.

views

Domanda del libro di testo

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

a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.

views

Domanda del libro di testo

a. Find the first four terms of the sequence of heights {hₙ}.

h₀ = 20,r = 0.5

views

Domanda del libro di testo

Explain why or why not

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

a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

views

Domanda del libro di testo

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

a. A series that converges must converge absolutely.

views