******************************************************************************************
Please Upvote the answer as it matters to me a lot :)
*****************************************************************************************
As HOMEWORKLIB RULES expert answering guidelines,Experts are supposed to
answer only certain number of questions/sub-parts in a post.Please
raise the remaining as a new question as HOMEWORKLIB RULES
guidelines.
******************************************************************************************
6. Use Mathematical Induction to show that (21 - 1)(2i+1) n for all integers n > 1. 2n +1 (5 marks) i=1
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
.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
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
Use the Principle of Mathematical Induction to prove that (2i+3) = n(n + 4) for all n > 1.
(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.
Problem 5.1.3. Prove by induction on n that (1+ n < n for every integer n > 3.
Use mathematical induction to show that when n is an exact power of 2, the solution of the recurrence: { if n 2 2 T(n) for k> 1 if n 2 T(n) 2T(n/2) is T(n) n log
n! 5. Let an On+1 <1 for all n. (1Show that an (2) Use (1) to show that {an} decreases. (3) Is {an} convergent?
(2) Prove by induction that for all integers n > 2. Hint: 2n-1-2n2,