1.
Which of the following is not always a step in mathematical induction?
A.
Prove true for some integer
n
.
B.
Prove true for
n
+ 1.
C.
Assume true for a positive integer
k
.
D.
Add (
k
+ 1)
2
to each side.
Hint
2.
Which of the following is a counterexample to the statement 8
k
– 1 = 9
r
for all
k
, where
k
and
r
are integers?
A.
(8
4
– 1) ÷ 9 = 455
B.
8
3
– 1 = 511, 511 ÷ 9 = 56.8
C.
8 = 9 – 1
D.
8(8
k
) = 8(9
r
+ 1)
Hint
3.
Which statement is a valid counterexample to the statement 6
n
+ 6
n
is divisible by 12?
A.
1
B.
4
C.
2
D.
3
Hint
4.
Is it true that 10
n
– 1 is divisible by 9?
A.
yes, for all integers
n
B.
no, since 10
5
– 1 = 49
C.
no, since 10
5
– 1 = 99999
D.
no, since 10
5
– 1 = 100001
Hint
5.
In induction, the statement is proven after which step?
A.
after
n
=
k
+ 1 is proven
B.
after
n
= 1 is proven
C.
after we assume true for
n
= 1
D.
after (
k
+ 1)
2
is added to each side.
Hint