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1. Prove by mathematical induction that for all positive integral values of n, 2=1 =- (n + 1)(2n + 1). Hence or otherwise, evaluate E=2(r-1) 1 n 2. Using mathematical induction, prove that 2"=1 4r2-1 for all positive integral values of 2n+1 n.

1. Prove by mathematical induction that for all positive integral values of n, 2=1 =- (n + 1)(2n + 1). Hence or otherwise, evaluate E=2(r-1) 1 n 2. Using mathematical induction, prove that 2"=1 4r2-1 for all positive integral values of 2n+1 n.

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...
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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
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