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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.3.5a
Chapter 10, Problem 10.3.5a

Find the first term a and the ratio r of each geometric series.


a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ

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Identify the general form of a geometric series, which is given by the sum from k = 0 to infinity of \( a \times r^k \), where \( a \) is the first term and \( r \) is the common ratio.
Compare the given series \( \sum_{k=0}^\infty \left( \frac{2}{3} \times \left( \frac{1}{5} \right)^k \right) \) to the general form to determine the first term \( a \) and the ratio \( r \).
Note that the first term \( a \) corresponds to the term when \( k = 0 \), which is \( \frac{2}{3} \times \left( \frac{1}{5} \right)^0 = \frac{2}{3} \times 1 = \frac{2}{3} \).
Identify the common ratio \( r \) as the factor raised to the power \( k \), which is \( \frac{1}{5} \).
Summarize that the first term \( a = \frac{2}{3} \) and the common ratio \( r = \frac{1}{5} \).

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Key Concepts

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

Geometric Series Definition

A geometric series is a sum of terms where each term after the first is found by multiplying the previous term by a constant ratio r. It can be written as a + ar + ar² + ar³ + ... , where a is the first term and r is the common ratio.
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Geometric Series

Identifying the First Term (a)

The first term a of a geometric series is the initial term when the index k is zero. In the series ∑ from k=0 to ∞ of a × r^k, the first term is simply the coefficient multiplied by r raised to the zero power, which equals a.
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The First Derivative Test: Finding Local Extrema Example 4

Common Ratio (r)

The common ratio r is the factor by which each term is multiplied to get the next term. It is the base of the exponent k in the series expression a × r^k, and it determines the behavior and convergence of the series.
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Related Practice
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.


a.Write the first five terms of the sequence {Bₙ}.

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Textbook Question

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.


a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.


∑ (k = 0 to ∞) (–2/7)ᵏ

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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.


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


h₀ = 30,r = 0.25

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Textbook Question

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.

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.


{1, 3, 9, 27, 81, ......}

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Textbook Question

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, …}.