Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
By dragging statements from the left column to the right column below, give a proof by induction of the following statement:   For all n≥1, 1^2+2^2+⋯+n^2=n(n+1)(2n+1) / 6

By dragging statements from the left column to the right column below, give a proof by induction of the following statement:   For all n≥1, 1^2+2^2+⋯+n^2=n(n+1)(2n+1) / 6

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

By dragging statements from the left column to the right column below, give a proof by induction of the following statement:

 
For all n≥1, 1^2+2^2+⋯+n^2=n(n+1)(2n+1) / 6
 
 
 
So 1² + 2² + ... + (k + 1)² k(k + 1)(2k + 1) 6 k(k + 1)(2k + 1) + (k + 1)² 6 (k+ 1)((k+ 1) + 1)(2(k + 1) + 1) 6 + (k + 1)² Note that 1² + 2² + ... + (k + 1)² = (1² + 2² + ... + k²) + (k+ 1)². Then 1² + 2² + ... + (k + 1)² = = k(k + 1)(2k + 1) 6 by the inductive hypothesis. Thus P(k) is true for all k. Now assume that P(k + 1) is true for all k. Thus P(k+ 1) is true.
Transcribed Image Text:So 1² + 2² + ... + (k + 1)² k(k + 1)(2k + 1) 6 k(k + 1)(2k + 1) + (k + 1)² 6 (k+ 1)((k+ 1) + 1)(2(k + 1) + 1) 6 + (k + 1)² Note that 1² + 2² + ... + (k + 1)² = (1² + 2² + ... + k²) + (k+ 1)². Then 1² + 2² + ... + (k + 1)² = = k(k + 1)(2k + 1) 6 by the inductive hypothesis. Thus P(k) is true for all k. Now assume that P(k + 1) is true for all k. Thus P(k+ 1) is true.
The correct proof will use 8 of the statements below. Statements to choose from: Note that 1² "1² + 2² + = = Let P(n) be the statement, + n² = (1)(1+1)(2.1+1) 6 1² + 2² + ... + k² n(n + 1)(2n + 1) „ 6 Then k(k + 1)(2k + 1) 6 Now assume that P(k) is true for an arbitrary integer k > 1. Then we see that 1² +2²2 + ... + (k + 1)² k(k + 1)(2k + 1) 6 k(k + 1)(2k + 1) + (k + 1)² 6(k+ 1)² 6 6 2k³ + 3k² + k 6k² + 12k +6 + 6 6 2k³ +9k² + 13k + 6 6 (k+ 1)(k+ 2)(2k + 3 6 (k+ 1)((k+ 1) + 1)(2(k + 1) + 1) 6 Therefore, by the Principle of Mathematical Induction, P(n) is true for all n ≥ 1.
Transcribed Image Text:The correct proof will use 8 of the statements below. Statements to choose from: Note that 1² "1² + 2² + = = Let P(n) be the statement, + n² = (1)(1+1)(2.1+1) 6 1² + 2² + ... + k² n(n + 1)(2n + 1) „ 6 Then k(k + 1)(2k + 1) 6 Now assume that P(k) is true for an arbitrary integer k > 1. Then we see that 1² +2²2 + ... + (k + 1)² k(k + 1)(2k + 1) 6 k(k + 1)(2k + 1) + (k + 1)² 6(k+ 1)² 6 6 2k³ + 3k² + k 6k² + 12k +6 + 6 6 2k³ +9k² + 13k + 6 6 (k+ 1)(k+ 2)(2k + 3 6 (k+ 1)((k+ 1) + 1)(2(k + 1) + 1) 6 Therefore, by the Principle of Mathematical Induction, P(n) is true for all n ≥ 1.
Expert Solution
steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS