1. Prove or disprove.
A. The difference of any two odd integers is odd.
B. An odd integer minus an even integer is odd.
2. Given (i=1)∑n [(2i-1)=n2]
A. Write P(n) without using the sigma (∑) notation.
B. Do a proof by mathematical induction using the 4 steps below to show that the statement is true for every positive integer n:
Hint: use the weak principle of mathematical induction.
Show that the base step is true
What is the inductive hypothesis?
What do we have to show?
Proof proper:
3. Do a proof by mathematical induction using the 4 steps below to show that the following statement is true for every positive integer n:
Hint: use the weak principle of mathematical induction.
5+10+15+...+5n =(5n(n+1))/2
Show that the base step is true
What is the inductive hypothesis?
What do we have to show?
Proof proper (Justify each step):
4. Prove using mathematical induction that 2^(3n) - 1 is divisible by 7 for all n ∈N.
a. Show that the base step is true
b. What is the inductive hypothesis?
c. What do we have to show?
d. Proof proper (Justify each step):