Homework Help is Here – Start Your Trial Now!
Math
Algebra
3. Given that n is a positive integer, prove by induction that, the 7"+ 5 is divisible by 6. Guess a common factor of 5" + 3 for all positive integers n and prove your guess by induction.

3. Given that n is a positive integer, prove by induction that, the 7"+ 5 is divisible by 6. Guess a common factor of 5" + 3 for all positive integers n and prove your guess by induction.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
See similar textbooks
icon
Related questions
Question

Solve Q3, 4 explaining detailly each step

MATHEMATICAL INDUCTION 1. Prove by mathematical induction that for all positive integral values of n, (1+ 1)(2n + 1). Hence or otherwise, evaluate E"=2r²(r –1) Lr=1 3D2 1 2. Using mathematical induction, prove that 2"=1 for all positive integral values of 4r 2 - 1 2n+1 n. 3. Given thatn is a positive integer, prove by induction that, the 7"+ 5 is divisible by 6. Guess a common factor of 5"+ 3 for all positive integers n and prove your guess by induction. 4. Prove by mathematical induction that for all positive integer n, 2n(n+2) E=1 2r(r + 1) = 5. (1) One of the statements that follow is true and the other is false. Prove the true statement and give a counter example to disprove the false statement. a. For all two dimensional vectors a, b, c, a.b = a.c →b = c b. For all positive real numbers a, b a+b > Vab (ii)Prove, by mathematical induction that r(2r + 1) = -(n+1)(4n+5) 6. a. Prove, by mathematical induction, that (4r + 3) = 2n2 + 5n b. Prove, by mathematical induction that /2 is not a rational number (you may assume that the square of an odd integer is always odd). 7. prove by mathematical induction that , 8. Prove by mathematical induction that 5"- 4n - 1 is divisible by 16, for all positive integers r (r+1) n+1 n. 9. Prove by mathematical induction that 8"-7n +6 is a multipie of 7, n e Z* n(n+1)(4n--1) 10. Prove by induction that =1r(r - 1) : 12 11. Prove by use of mathematical induction, that , r 12. DProve by mathematical induction that: 9"-1 is divisible by 8, for ail positive integers n. ii) (n + 1)2 3. Determine the value(s) of x for which o K+1/ 4. 13. Prove by mathematical induction, that: r(r + 1) 1. = n (n+1) (n+2), for all positive 3. integers n. 14. Prove by mathematical induction that: 2-1(2 + 3r) =-(3n + 7), for all positive integers n. Prove by mathematical induction that 3 + 7 is divisible by 8 for all integers n > 0
Transcribed Image Text:MATHEMATICAL INDUCTION 1. Prove by mathematical induction that for all positive integral values of n, (1+ 1)(2n + 1). Hence or otherwise, evaluate E"=2r²(r –1) Lr=1 3D2 1 2. Using mathematical induction, prove that 2"=1 for all positive integral values of 4r 2 - 1 2n+1 n. 3. Given thatn is a positive integer, prove by induction that, the 7"+ 5 is divisible by 6. Guess a common factor of 5"+ 3 for all positive integers n and prove your guess by induction. 4. Prove by mathematical induction that for all positive integer n, 2n(n+2) E=1 2r(r + 1) = 5. (1) One of the statements that follow is true and the other is false. Prove the true statement and give a counter example to disprove the false statement. a. For all two dimensional vectors a, b, c, a.b = a.c →b = c b. For all positive real numbers a, b a+b > Vab (ii)Prove, by mathematical induction that r(2r + 1) = -(n+1)(4n+5) 6. a. Prove, by mathematical induction, that (4r + 3) = 2n2 + 5n b. Prove, by mathematical induction that /2 is not a rational number (you may assume that the square of an odd integer is always odd). 7. prove by mathematical induction that , 8. Prove by mathematical induction that 5"- 4n - 1 is divisible by 16, for all positive integers r (r+1) n+1 n. 9. Prove by mathematical induction that 8"-7n +6 is a multipie of 7, n e Z* n(n+1)(4n--1) 10. Prove by induction that =1r(r - 1) : 12 11. Prove by use of mathematical induction, that , r 12. DProve by mathematical induction that: 9"-1 is divisible by 8, for ail positive integers n. ii) (n + 1)2 3. Determine the value(s) of x for which o K+1/ 4. 13. Prove by mathematical induction, that: r(r + 1) 1. = n (n+1) (n+2), for all positive 3. integers n. 14. Prove by mathematical induction that: 2-1(2 + 3r) =-(3n + 7), for all positive integers n. Prove by mathematical induction that 3 + 7 is divisible by 8 for all integers n > 0
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Linear Equations
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 and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS