/*! 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 Find the common ratio, \(r,\) fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 3

Find the common ratio, \(r,\) for each geometric sequence. $$1,2,4,8, \dots$$

Short Answer

Expert verified
The common ratio, \(r\), for the given geometric sequence \(1, 2, 4, 8, \dots\) is \(2\).

Step by step solution

01

Identify the terms of the sequence

In the given sequence, we can identify the terms as follows: First term: \(a_1 = 1\) Second term: \(a_2 = 2\) Third term: \(a_3 = 4\) Fourth term: \(a_4 = 8\) and so on.
02

Use the geometric sequence formula to find the common ratio

To express the geometric sequence formula, we can use the expression: \(a_n = a_1 \cdot r^{n-1}\) Where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
03

Divide the consecutive terms

To find the common ratio, \(r\), we can simply divide the consecutive terms of the sequence. Since the sequence is geometric, the division result should be constant for any consecutive terms: Divide the second term by the first term: \(\frac{a_2}{a_1} = \frac{2}{1} = 2\) Divide the third term by the second term: \(\frac{a_3}{a_2} = \frac{4}{2} = 2\) Divide the fourth term by the third term: \(\frac{a_4}{a_3} = \frac{8}{4} = 2\) Since all the results are the same, we have found the common ratio, \(r = 2\). To conclude, the common ratio, \(r\), for the given geometric sequence \(1, 2, 4, 8, \dots\) is \(2\).

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
The common ratio in a geometric sequence is the constant factor that each term is multiplied by to get to the next term. This ratio is consistent throughout the sequence. To find the common ratio, you need to divide any term in the sequence by the previous term. By doing this, you'll find the multiplier that connects the terms. In our example sequence of \(1, 2, 4, 8, \dots\), the common ratio \(r\) was found by dividing consecutive terms as follows:
  • \(\frac{2}{1} = 2\)
  • \(\frac{4}{2} = 2\)
  • \(\frac{8}{4} = 2\)
As you can see, the common ratio here is \(r = 2\), signaling that each term is double the preceding term.
Consecutive Terms
Consecutive terms in a geometric sequence are the terms that follow one after another. In sequences, these terms are related by the common ratio. Identifying consecutive terms is important as it allows you to calculate the common ratio. For any geometric sequence:
  • First Term (\(a_1\)): This is the initial term from which the sequence starts.
  • Second Term (\(a_2\)): This follows the first term and can be reached by multiplying \(a_1\) by \(r\).
  • Third Term (\(a_3\)), Fourth Term (\(a_4\)), and so on: Each term follows this multiplication pattern with \(r\).
By understanding how consecutive terms are derived, students can better grasp how changes in the sequence follow logic and pattern rather than random values.
Geometric Sequence Formula
The geometric sequence formula is used to describe the relationship between the terms of a geometric sequence. The formula is:\[ a_n = a_1 \cdot r^{n-1} \]where:
  • \(a_n\) is the nth term in the sequence.
  • \(a_1\) is the first term of the sequence.
  • \(r\) is the common ratio for the sequence.
  • \(n\) is the position of the term in the sequence.
This formula allows you to find any term in the sequence without listing all the preceding terms if you know the first term and the common ratio. In our example with the sequence \(1, 2, 4, 8, \dots\), if you want to find the fifth term (\(a_5\)), you simply use the formula:\[ a_5 = 1 \cdot 2^{5-1} = 16 \]Utilizing this formula is a powerful way to analyze and predict geometric sequences.

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 sum of the terms of each geometric sequence. $$\sum_{i=1}^{4}(-18)\left(-\frac{2}{3}\right)^{i}$$

Use the binomial theorem to expand each expression. $$(2 k+1)^{4}$$

Find the indicated term of each binomial expansion. $$(3 x-2)^{6} ; \text { fifth term }$$

A ball is dropped from a height of 16 ft. Each time the ball bounces it rebounds to \(\frac{3}{4}\) of its previous height. a) Find the height the ball reaches after the fourth bounce. b) Find the total vertical distance the ball has traveled when it comes to rest.

Use the binomial theorem to expand each expression. $$(4 c-3 d)^{4}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete 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.