n. Prove by mathematical induction that 3 + 7 is divisible by 8 for all integers n >0(

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
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Solve Q14, 15 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
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