/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 What is meant by the ratio of a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 3

What is meant by the ratio of a geometric series?

Short Answer

Expert verified
Answer: The ratio of a geometric series, denoted as 'r,' is the constant factor by which each term is multiplied to get to the next term. It indicates the relationship between consecutive terms in the series. To calculate the common ratio (r) in a given geometric series, follow these steps: 1. Choose any two consecutive terms in the series. 2. Divide the second term by the first term. 3. The result obtained is the common ratio (r) of the geometric series.

Step by step solution

01

Introduction to Geometric Series

A geometric series is a mathematical sequence of numbers in which each term after the first one is determined by multiplying the previous term by a fixed, non-zero constant, called the common ratio (denoted as r).
02

Formula of Geometric Series

To better understand the ratio in a geometric series, let's first look at the formula for a geometric series. A general geometric sequence can be represented as: a_n = a_1 * r^(n-1) where 'a_n' is the nth term in the series, 'a_1' is the first term, 'r' is the common ratio, and 'n' is the position of the term in the series.
03

Ratio in a Geometric Series

The ratio (r) in a geometric series is the constant factor by which each term is multiplied to get to the next term. It indicates the relationship between consecutive terms in the series. If r > 1, the series increases, and if 0 < r < 1, the series decreases. For example, consider the geometric series: 2, 10, 50, 250, ... To find the ratio, you can divide any term by the previous term: r = a_2 / a_1 = a_3 / a_2 = a_4 / a_3 = ... In this example: r = 10/2 = 50/10 = 250/50 = 5 This means that each term in this geometric series is multiplied by 5 to get the next term.
04

Calculating the Ratio in a Geometric Series

To calculate the common ratio (r) in a given geometric series, follow these steps: 1. Choose any two consecutive terms in the given series. 2. Divide the second term by the first term. 3. The result obtained is the common ratio (r) of the geometric series. For example, if the given geometric series is: 3, 6, 12, 24, ... 1. Choose any two consecutive terms, such as 6 and 12. 2. Divide the second term (12) by the first term (6): 12/6 = 2. 3. The common ratio (r) for this geometric series is 2. In this way, you can find the ratio of any geometric series.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

$$\text {Evaluate each series or state that it diverges.}$$ $$\sum_{k=1}^{\infty}\left(\sin ^{-1}(1 / k)-\sin ^{-1}(1 /(k+1))\right)$$

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$

$$\text {Evaluate each series or state that it diverges.}$$ $$\sum_{k=1}^{\infty} \frac{\pi^{k}}{e^{k+1}}$$

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. Radioactive decay A material transmutes \(50 \%\) of its mass to another element every 10 years due to radioactive decay. Let \(M_{n}\) be the mass of the radioactive material at the end of the \(n\) th decade, where the initial mass of the material is \(M_{0}=20 \mathrm{g}\)

In Section \(8.3,\) we established that the geometric series \(\sum r^{k}\) converges provided \(|r| < 1\). Notice that if \(-1 < r<0,\) the geometric series is also an alternating series. Use the Alternating Series Test to show that for \(-1 < r <0\), the series \(\sum r^{k}\) converges.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.