/*! 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 11 Find the common ratio for each g... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 11

Find the common ratio for each geometric sequence. $$10,20,40,80, \dots$$

Short Answer

Expert verified
The common ratio is 2.

Step by step solution

01

Identify Consecutive Terms

Select any two consecutive terms in the geometric sequence. For simplicity, choose the first term (10) and the second term (20).
02

Apply the Common Ratio Formula

The common ratio of a geometric sequence can be found by dividing any term by its preceding term. Use the formula: \[ r = \frac{a_{n+1}}{a_n} \] where \(a_{n+1}\) and \(a_n\) are consecutive terms.
03

Substitute the Terms

Substitute the first term (10) and the second term (20) into the formula: \[ r = \frac{20}{10} \]
04

Simplify the Ratio

Divide 20 by 10 to simplify the ratio: \[ r = 2 \]

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 a geometric sequence, the common ratio is a constant factor between consecutive terms. This ratio helps determine how the sequence progresses. To find the common ratio, use the formula: \ \[ r = \frac{a_{n+1}}{a_n} \] \ where \(a_{n+1}\) is the term that comes after \(a_n\). Simply pick any two consecutive terms from the sequence and divide the later term by the earlier one. For example, in the sequence \(10, 20, 40, 80, \ \ ...\), \( r \) is calculated as follows: \ \[ r = \frac{20}{10} = 2 \]. \ This shows that each term is twice its preceding term. Understanding the common ratio is crucial for comprehending how geometric sequences behave and grow.
Consecutive Terms
Consecutive terms in a geometric sequence are two terms that follow one another directly. For example, in the sequence \(10, 20, 40, 80\), the numbers 10 and 20 are consecutive terms. Each term is obtained by multiplying the previous term by the common ratio. Knowing this helps us understand the relationship between terms and determine the common ratio effectively. \ Recognizing consecutive terms is the first step in solving problems related to geometric sequences. \ This foundational step ensures that we apply the correct terms when using the common ratio formula.
Simplification of Ratios
Simplifying ratios is an essential mathematical skill, especially in geometric sequences. It involves reducing a fraction to its simplest form. For example, in the sequence \(10, 20, 40, 80\), the common ratio is found using the formula: \ \[ r = \frac{20}{10} \] \ Here, \(r\) simplifies to \(2\) after dividing 20 by 10. Simplifying ratios helps us clearly understand the growth factor in the sequence without dealing with unnecessarily complicated fractions. \ Remember, always divide the terms to their lowest terms for clarity and simplicity in calculations. This ensures easier handling and understanding of the sequence's pattern and future calculations.

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 formula for \(S_{n}\) to find the indicated sum for each geometric series. $$S_{9} \text { for } 6+12+24+\cdots$$

Find the indicated term for each binomial expression. $$\text { Middle, }\left(2 u+3 v^{2}\right)^{10}$$

Find the first five terms of each sequence; then find \(S_{5}\). $$ a_{n}=\frac{1}{2^{n}} \log 1000^{n} $$

Find fraction notation for each repeating decimal. $$0.5555 \ldots$$

Solve. $$|2 x+5|<6$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.