6. Use Mathematical Induction to show that (21 - 1)(2i+1) n for all integers n > 1. 2n +1 (5 marks) i=1
Problem 44) Prove: n!> 2" for n24. Problem 45) Prove by induction: For n>0·AT- i=1
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4) +...+(n-1).n [9 points) 3
Prove by induction that 1 1 15 15 + 1 35 35 + ... + = n 2n + 1 for every n > 1 4n2
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
(a) Use mathematical induction to prove that for all integers n > 6, 3" <n! Show all your work. (b) Let S be the subset of the set of ordered pairs of integers defined recursively by: Basis Step: (0,0) ES, Recursive Step: If (a, b) ES, then (a +2,5+3) ES and (a +3,+2) ES. Use structural induction to prove that 5 (a + b), whenever (a, b) E S. Show all your work.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
Use the Principle of mathematical induction to prove 2. Use the Principle of Mathematical Induction to prove: Lemma. Let n E N with n > 2, and let al, aa-.., an E Z all be nonzero. If gcd(ai ,aj) = 1 for all i fj, then gcd(aia2an-1,an)1. 1, a2,, an
Problem 7: Prove that for all integers n > 2, n+1 n 10-11 - n n +
8. Use mathematical induction to prove that n + + 7n 15 3 5 is an integer for all integers n > 0.