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 10.2.73b

Chapter 10, Problem 10.2.73b

{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}.

Verified step by step guidance

Understand the problem setup: James deposits \$100 at the beginning of each month, and the account earns 0.75% interest monthly. The interest is added first, then the deposit is made each month.

Define the variables clearly: Let $B_n$ be the balance after the $n^{th}$ deposit. Given $B_0 = 0$, we want to express $B_n$ in terms of $B_{n-1}$.

Express the interest accumulation: Before the $n^{th}$ deposit, the balance $B_{n-1}$ earns 0.75% interest. This means the balance grows by a factor of $1 + 0.0075$ (since 0.75% = 0.0075 in decimal). So, the balance after interest but before deposit is $B_{n-1} \times (1 + 0.0075)$.

Add the monthly deposit: After the interest is added, James deposits \$100. So, the new balance after the $n^{th}$ deposit is $B_n = B_{n-1} \(\times$ (1 + 0.0075) + 100\).

Write the recurrence relation explicitly: \[ B_n = 1.0075 \times B_{n-1} + 100, \quad \text{with} \quad B_0 = 0. \]

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Key Concepts

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

Recurrence Relations

A recurrence relation defines each term of a sequence using previous terms. In this problem, the balance after each payment depends on the previous balance and the new deposit, making it essential to express Bₙ in terms of Bₙ₋₁. Understanding how to set up such relations helps model the growth of the savings over time.

Compound Interest

Compound interest means interest is earned on both the initial principal and the accumulated interest from previous periods. Here, the account earns 0.75% interest monthly, which is added before the deposit. Recognizing how interest compounds monthly is crucial to correctly formulating the recurrence.

Sequence and Series in Financial Contexts

Sequences represent the balance after each payment, and understanding their behavior helps analyze savings growth. Financial sequences often involve regular deposits and interest accumulation, requiring knowledge of how to combine arithmetic (deposits) and geometric (interest) components in the recurrence.

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Textbook Question

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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Textbook Question

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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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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Textbook Question