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 10.2.73a

Chapter 10, Problem 10.2.73a

{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.

a.Write the first five terms of 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 sequence {Bₙ} where B₀ = 0 represents the balance before any deposits or interest. For each month n, the balance after the nth deposit is Bₙ.

Express the recursive formula for Bₙ: The balance after the nth payment is the previous balance Bₙ₋₁ grown by 0.75% interest, plus the new \$100 deposit. Mathematically, this is: $B_{n} = B_{n-1} \times (1 + 0.0075) + 100$

Calculate the first five terms step-by-step using the recursive formula: - $B_0 = 0$ - $B_1 = B_0 \times 1.0075 + 100$ - $B_2 = B_1 \times 1.0075 + 100$ - $B_3 = B_2 \times 1.0075 + 100$ - $B_4 = B_3 \times 1.0075 + 100$ - $B_5 = B_4 \times 1.0075 + 100$

Write out each term explicitly by substituting the previous term's value, showing how the balance grows month by month with interest and deposits.

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Key Concepts

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

Recursive Sequences

A recursive sequence defines each term based on the previous term(s). In this problem, the balance after each payment depends on the previous balance plus interest and the new deposit. Understanding how to express Bₙ in terms of Bₙ₋₁ is essential to write out the sequence terms.

Compound Interest

Compound interest means interest is earned on both the initial principal and the accumulated interest from previous periods. Here, interest is added monthly at 0.75%, so the balance grows by multiplying the previous balance by 1.0075 before adding the new deposit.

Sequence Initialization and Notation

The sequence starts with B₀ = 0, representing the initial balance before any deposits. Correctly interpreting the timing of interest addition and deposits is crucial to accurately calculate each term Bₙ, especially since interest is added before the deposit each month.

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Textbook Question

88–89. Binary numbers

Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite series

b₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯

89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.

a. Find the base-10 representation of the binary number 0.001̅1.

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Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)

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Textbook Question

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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Textbook Question