/*! 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 51 Find \(r\) for each infinite geo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 51

Find \(r\) for each infinite geometric sequence. Identify any whose sum does not converge. $$-48,-24,-12,-6, \dots$$

Short Answer

Expert verified
The common ratio \(r\) is \(\frac{1}{2}\). The series converges.

Step by step solution

01

Identify the first term

The first term of the infinite geometric sequence, denoted by \(a_1\), is \(-48\).
02

Determine the common ratio

To find the common ratio \(r\) of the geometric sequence, divide the second term by the first term: \( r = \frac{-24}{-48} = \frac{1}{2} \).
03

Check for convergence

For the sum of an infinite geometric sequence to converge, \(|r| < 1\) must be true. Here, \(|r| = \left| \frac{1}{2} \right| = \frac{1}{2} < 1\), so the series converges.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
In an infinite geometric sequence, the common ratio is a key element. It tells us how each term is derived from the previous one. In the given sequence, -48, -24, -12, -6, ..., the common ratio is determined by dividing any term by the previous term. For example, the second term (-24) divided by the first term (-48) results in a common ratio of \( r = \frac{-24}{-48} = \frac{1}{2} \).

This constant ratio \( \frac{1}{2} \) means that each subsequent term is half the preceding term. This constant multiplicative factor is what defines the pattern of a geometric sequence. Understanding this ratio helps to predict future terms and analyze the overall behavior of the sequence.
Convergence
Convergence in a geometric series refers to whether the sum of all the terms will approach a finite number as the sequence progresses to infinity. For a geometric series to converge, the common ratio \( r \) must satisfy a specific condition: the absolute value of \( r \) should be less than one \(|r| < 1\).

In the sequence -48, -24, -12, -6, ..., the common ratio was calculated as \( \frac{1}{2} \). Since \( \left| \frac{1}{2} \right| < 1 \), this sequence's sum does indeed converge. Convergence is an important concept because it indicates stability in the progression, meaning the series does not grow indefinitely but settles into a limit instead.
Geometric Series
A geometric series is simply the sum of the terms in a geometric sequence. When dealing with infinite geometric series, we often seek whether they converge to a certain sum.

For a geometric series with first term \( a_1 \) and common ratio \( r \), its sum can be calculated using a special formula if \(|r| < 1\):\[S = \frac{a_1}{1 - r}\]In the sequence -48, -24, -12, -6, ..., the first term \( a_1 \) is -48, and \( r \) is \( \frac{1}{2} \). Therefore, this series converges to:\[S = \frac{-48}{1 - \frac{1}{2}} = \frac{-48}{\frac{1}{2}} = -96.\]This formula empowers us to find the limit that the infinite series approaches, facilitating deeper understanding of sequences' behaviors.

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

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. $$a_{n}=4.2 n+9.73$$

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{1}=10, a_{10}=5.5$$

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{2}=9, a_{4}=13$$

Find the sum of each series. $$\sum_{i=1}^{19}(-3-4 k)$$

Find the sum of all the integers from \(-8\) to 30.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.