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 10.2.73c

Chapter 10, Problem 10.2.73c

{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.

c.How many months are needed to reach a balance of $5000?

Verified step by step guidance

Recognize that the balance after the nᵗʰ payment, $B_n$, follows a recurrence relation because each month the bank first adds interest to the current balance, then James deposits \$100.

Express the recurrence relation as: \[B_n = B_{n-1} \times (1 + 0.0075) + 100,\] where $0.0075$ is the monthly interest rate (0.75%).

Identify that this is a non-homogeneous linear recurrence relation, which can be solved using the formula for the sum of a geometric series or by recognizing it as a future value of an annuity with monthly compounding.

Write the explicit formula for $B_n$ after $n$ months: \[B_n = 100 \times \frac{(1 + 0.0075)^n - 1}{0.0075}.\] This formula comes from summing the compounded deposits over $n$ months.

Set $B_n = 5000$ and solve for $n$: \[5000 = 100 \times \frac{(1 + 0.0075)^n - 1}{0.0075}.\] Rearrange to isolate $(1 + 0.0075)^n$ and then take the natural logarithm to solve for $n$.

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Key Concepts

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

Compound Interest with Periodic Deposits

Compound interest means interest is earned on both the initial principal and the accumulated interest from previous periods. When regular deposits are made, each deposit grows with interest over time, creating a series of compounded amounts. Understanding how interest is applied before or after deposits is crucial for accurate calculations.

Recurrence Relation for Account Balance

The balance after each month can be expressed using a recurrence relation: the new balance equals the previous balance plus interest, then plus the new deposit. This relation helps model the growth of the account step-by-step and is essential for finding the balance after n months or determining n for a target balance.

Solving for Number of Periods in Annuity Problems

To find how many months are needed to reach a target balance, one uses formulas for the future value of an annuity with compound interest. This involves solving for the number of periods n in the equation that relates periodic deposits, interest rate, and final balance, often requiring logarithms or iterative methods.

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Related Practice

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.

c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

Textbook Question

Explain why or why not

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

c.If the terms of the sequence {aₙ} are positive and increasing, then the sequence of partial sums for the series∑⁽∞⁾ₖ₌₁aₖ diverges.

Textbook Question

87. Explain why or why not

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

c. If ∑ aₖ converges, then ∑ (aₖ + 0.0001) converges.

Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.

{-5, 5, -5, 5, ......}

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Textbook Question

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

c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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