Homework Help is Here – Start Your Trial Now!
Math
Calculus
Use the principle of mathematical induction to show that the statement is true for all natural numbers. + (2n)³ = 2n²(n + 1)² 23 +43 +63 + www

Use the principle of mathematical induction to show that the statement is true for all natural numbers. + (2n)³ = 2n²(n + 1)² 23 +43 +63 + www

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question
↓ Use the principle of mathematical induction to show that the statement is true for all natural numbers. 23 +43 +63 + ... + (2n)3 = 2n²(n + 1)² Let Pn denote the statement: 23+ 43 + 63 + ... + (2n)³ = 2n²(n + 1)². 2 ²()(C + 1)² Check that P₁ is true: 23 = 8 and Assume Pk is true: 23 +43 +63 + ... 23 +43 +63 +...+(2k)³ + To show that Pk+1 is true, add (2(k + 1))³ to both sides of Pk. ³ +(²( Completely factor the right-hand side. +(2 23 +43 +63 + ... + (2(k + 1))³ = = 2( 2( = 3 )² - 24³² ( = W )²( 2 |)²(K² + Thus P₁ is true. 3 ))³ = 2K² (K + 1)² + (2 (k 2 2+4( 2 + 4 3 1))³ ))
Transcribed Image Text:↓ Use the principle of mathematical induction to show that the statement is true for all natural numbers. 23 +43 +63 + ... + (2n)3 = 2n²(n + 1)² Let Pn denote the statement: 23+ 43 + 63 + ... + (2n)³ = 2n²(n + 1)². 2 ²()(C + 1)² Check that P₁ is true: 23 = 8 and Assume Pk is true: 23 +43 +63 + ... 23 +43 +63 +...+(2k)³ + To show that Pk+1 is true, add (2(k + 1))³ to both sides of Pk. ³ +(²( Completely factor the right-hand side. +(2 23 +43 +63 + ... + (2(k + 1))³ = = 2( 2( = 3 )² - 24³² ( = W )²( 2 |)²(K² + Thus P₁ is true. 3 ))³ = 2K² (K + 1)² + (2 (k 2 2+4( 2 + 4 3 1))³ ))
Completely factor the right-hand side. 23 + 43 + 6³ + ... + (2(k + 1))³ = 2( = 2([ = 2( Rewrite the right-hand side in the desired form. 2³ + 4³ + 6³ + ... + (2(k + 1))³ = DO ) ² ( K² ) ² ( K + + 2 + ) 2 So Pk+1 is true. We conclude by the principle of mathematical induction that Pn is true for all natural numbers n. Need Help? Read It
Transcribed Image Text:Completely factor the right-hand side. 23 + 43 + 6³ + ... + (2(k + 1))³ = 2( = 2([ = 2( Rewrite the right-hand side in the desired form. 2³ + 4³ + 6³ + ... + (2(k + 1))³ = DO ) ² ( K² ) ² ( K + + 2 + ) 2 So Pk+1 is true. We conclude by the principle of mathematical induction that Pn is true for all natural numbers n. Need Help? Read It
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS