Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.1.73b

Chapter 10, Problem 10.1.73b

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.

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.

Verified step by step guidance

Identify the type of sequence described. Since the material loses 50% of its mass every 10 years, the sequence represents exponential decay, which is a geometric sequence.

Determine the common ratio $ r $ of the geometric sequence. Because the material retains 50% of its mass each decade, the ratio is $ r = 0.5 $.

Write the general form of the explicit formula for a geometric sequence: $ M_n = M_0 \times r^n $, where $ M_0 $ is the initial mass and $ n $ is the number of decades.

Substitute the given initial mass $ M_0 = 20 $ grams and the common ratio $ r = 0.5 $ into the formula to get $ M_n = 20 \times (0.5)^n $.

Interpret the formula: $ M_n $ gives the mass of the radioactive material after $ n $ decades, showing how the mass decreases by half every 10 years.

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Key Concepts

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

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 an explicit formula for the nth term based on the initial value and the common ratio.

Explicit Formula for Sequences

An explicit formula expresses the nth term of a sequence directly in terms of n, without needing previous terms. For geometric sequences, the formula is Mₙ = M₀ * rⁿ, where M₀ is the initial term and r is the common ratio. This allows calculation of any term in the sequence efficiently.

Radioactive Decay Modeling

Radioactive decay describes how a substance decreases over time at a consistent fractional rate. Modeling decay with sequences involves applying the decay factor repeatedly over equal time intervals. This real-world context helps connect abstract sequence concepts to practical applications.

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Exponential Growth & Decay

Related Practice

Textbook Question

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)]

Textbook Question

{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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Textbook Question

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

b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.

41. ∑ (k = 1 to ∞) 1 / k⁶

Textbook Question

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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Textbook Question

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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Textbook Question

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, 2, 4, 8, 16, ......}

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