Explanation Given: The series of sketches starts with an equilateral triangle having sides of length 1 . In the following steps, equilateral triangles are constructed on each side of the preceding figure. The length of the sides of each new triangle is 1 3 the length of the sides of the preceding triangles. Develop a formula for the number of sides of the n t h figure. Use mathematical induction to prove the answer. Given that the series of sketches starts with an equilateral triangle having sides of length . In the following steps, equilateral triangles are constructed on each side of the preceding figure. The length of the sides of each new triangle is 1 3 the length of the sides of the preceding triangles. The first triangle has 3 sides. The second figure contains 12 sides. The third figure contains 48 sides. The fourth figure contains 128 sides of a triangle. So the series should be as follows: 3 , 12 , 48 , 192 , ... .The series can be rewritten as: 3 ( 4 ) 0 , 3 ( 4 ) 1 , 3 ( 4 ) 3 , 3 ( 4 ) 4 , ..........3 ( 4 ) n − 1 where n is a natural number. It can be rewritten as 3 [ 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 ] . The sum of the geometric series: 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 = 4 n − 1 4 − 1 = 4 n − 1 3 . Thereby our given series becomes: 3 [ 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 ] = 3 [ 4 n − 1 3 ] . On rewriting this equation we can obtain the statement which is to be proved, P ( n ) : 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 = 4 n − 1 3 By Proof of Induction, in the first step we have to prove P ( n ) is true for n = 1 . In the second step is we have to consider P ( n ) is true for n = k , where k is any value. It means P ( k ) is true. This is called Induction Hypothesis. We will now prove directly that P ( k ) ⇒ P ( k + 1 ) , which means P ( k + 1 ) is true by using the fact that P ( k ) is true. This is called Induction Step. The combination of these steps shows that P ( n ) is true for all values of n. Formula used: P ( n ) : 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 = 4 n − 1 3 . Calculation: Description P ( n ) : 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 = 4 n − 1 3 Step 1: To prove P ( 1 ) is true...

College Algebra with Modeling & Visualization (5th Edition)

5th Edition

ISBN: 9780321826138

Author: Gary K. Rockswold

Publisher: PEARSON

See similar textbooks

Videos

Question

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 8.5, Problem 30E

To determine

Formula for the number of sides of the $n^{t h}$ figure. Use mathematical induction to prove the answer.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.5, Problem 29E

Chapter 8.5, Problem 31E

Chapter 8 Solutions

College Algebra with Modeling & Visualization (5th Edition)

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Algebra and Trigonometry (6th Edition)
Algebra
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:9781305657960
Author:Joseph Gallian
Publisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:9780135163078
Author:Michael Sullivan
Publisher:PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:9780980232776
Author:Gilbert Strang
Publisher:Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:9780077836344
Author:Julie Miller, Donna Gerken
Publisher:McGraw-Hill Education

Algebra and Trigonometry (6th Edition)

Algebra

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:PEARSON

Contemporary Abstract Algebra

Algebra

ISBN:9781305657960

Author:Joseph Gallian

Publisher:Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:9781285463247

Author:David Poole

Publisher:Cengage Learning

Algebra And Trigonometry (11th Edition)

Algebra

ISBN:9780135163078

Author:Michael Sullivan

Publisher:PEARSON

Introduction to Linear Algebra, Fifth Edition

Algebra

ISBN:9780980232776

Author:Gilbert Strang

Publisher:Wellesley-Cambridge Press

College Algebra (Collegiate Math)

Algebra

ISBN:9780077836344

Author:Julie Miller, Donna Gerken

Publisher:McGraw-Hill Education

SEE MORE TEXTBOOKS

What are the Different Types of Triangles? | Don't Memorise; Author: Don't Memorise;https://www.youtube.com/watch?v=1k0G-Y41jRA;License: Standard YouTube License, CC-BY

Law of Sines AAS, ASA, SSA Ambiguous Case; Author: Mario's Math Tutoring;https://www.youtube.com/watch?v=FPVGb-yWj3s;License: Standard YouTube License, CC-BY

Introduction to Statistics..What are they? And, How Do I Know Which One to Choose?; Author: The Doctoral Journey;https://www.youtube.com/watch?v=HpyRybBEDQ0;License: Standard YouTube License, CC-BY

Triangles | Mathematics Grade 5 | Periwinkle; Author: Periwinkle;https://www.youtube.com/watch?v=zneP1Q7IjgQ;License: Standard YouTube License, CC-BY

What Are Descriptive Statistics And Inferential Statistics?; Author: Amour Learning;https://www.youtube.com/watch?v=MUyUaouisZE;License: Standard Youtube License

Formula for the number of sides of the n t h figure. Use mathematical induction to prove the answer.

Videos

Formula for the number of sides of the nth figure. Use mathematical induction to prove the answer.

Want to see the full answer?

Chapter 8 Solutions

Formula for the number of sides of the $n^{t h}$ figure. Use mathematical induction to prove the answer.