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COT 3100
Summer 2001
Quiz #4 (Solutions)
07/12/2001
Total 25 pts
1.
(7pts) Use induction to show that 5
n

1 is divisible by 4 for
n
=1, 2, …
Basis:
n
=1, LHS=5
1

1=4 is divisible by 4.
IH.
Assume that for
n
=
k
,
k
is some integer
k
≥
1, the proposition is true,
i. e. 5
k

1 =4
⋅
s
,
s
is any integer.
IS
. We need to prove that 5
k
+1

1 is divisible by 4, i.e. 5
k
+1

1 = 4
⋅
p
,
p
is any integer
.
5
k
+1

1 = 5 (5
k

1)+4 =5
⋅
4
⋅
s
+4 (by IH)
= 4 (5
⋅
s
+1) = 4
p
, where
p
= 4
⋅
s
+ 1, an integer.
2.
(8pts) Prove that 2
n
>
n
2
whenever
n
is an integer greater than 4.
Basis
.
n
= 5, 2
5
=32> 5
2
=25.
IH
. Assume for
n
=
k
, where
k
is some integer
k
>4, the inequality holds, i.e. 2
k
>
k
2
.
IS
. We need to prove that the inequality holds for
n
=
k
+1, i.e. 2
k
+1
>(
k
+1)
2
.
2
k
+1
= 2
⋅
2
k
> 2
⋅
k
2
(by IH)
=
k
2
+
k
2
>
k
2
+2
⋅
k
+1
=(
k
+1)
2
,
because
k
2
>2
⋅
k
+1
for
k
>4. The reason why it is true can be implied from the fact
that the quadratic polynomial
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