MTH1051 Week 5 Assignment 1 Complete Solution

Question: #1050

MTH1051

Chapter 5

True / False Questions pg 156

    The statement “∑_(i=1)^n〖(2i-1)=n^2 〗 for every n € N” is the type of statement that can be proved by mathematical induction.

    The statement “2^3n-1is divisible by 7 for every n € N” is the type of statement that can be proved using mathematical induction.

    The statement 2^x> x^2 for every real number x ≥ 5” is the type of statement that can be proved using mathematical induction.

    The statement “(1┤+1/2)" ≥ 1 + n/2 for every integer n” is the type of statement that can be proved using one application of mathematical induction.

    The second step in the Principle of Mathematical Induction is called the induction hypothesis.

    Mathematical induction is not necessary if a high-speed computer can be used to check the first 100,000 cases of a statement about all n € N.

    The strong form of the Principle of Mathematical Induction differs from the usual form only in the statement of the induction hypothesis.

Pg 156

4) Use mathematical induction to prove the truth of each of the following assertions for all n ≥ 1

    b) n^3+ 〖(n+1)〗^3+ 〖(n+2)〗^3 is divisible by 9.

    d) 8^n- 3^n is divisible by 5

    g) n^3+5n is divisible by 6

5)
    b) Prove by mathematical induction that 1^3+ 2^3+⋯+ n^3= (n^2 〖(n+1)〗^2)/4 for any natural number n.

    Use mathematical induction to establish the truth of each of the following statements for all n ≥ 1.

    1 + 2 + 2^2+ 2^3+⋯+2^n= 2^(n+1)-1

e) 1/(1*2)+ 1/(2*3)+ 1/(3*4)+⋯+ 1/n(n+1) = n/(n+1)

    Use mathematical induction to establish each of the following formulas.

    ∑_(i=1)^n〖i^2/((2i-1)(2i+1))= (n(n+1))/(2(2n+1))〗

    Use mathematical induction to establish each of the following inequalities.

    〖(1+1/2)〗^n ≥1+ n/2, for n € N.

g) 1/1^2 + 1/2^2 + 1/3^2 +⋯+ 1/n^2

Workbook pg 78

Prove each proposition using the principle of mathematical induction

42) 1^3+ 2^3+ 3^3+⋯+ n^3= (n^2 〖(n+1)〗^2)/4

44) 1 + 10 + 100 +⋯ + 〖10〗^n= (〖10〗^(n+1)-1)/9