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Answers, Assignment #9
Exercises 1.1.
1. Let
S
be the set of integers for which the statement is true.
Since 1 =
1
·
2
2
=1
,
1
∈
S.
Assume that the statement is true for some positive integer
n
=
k
Then, for
n
=
k
+1,
1+2+
···
+
k
+(
k
+1) = [1+2+
+
k
]+(
k
+1)=
k
(
k
+1)
2
k
=
k
(
k
+ 1) + 2(
k
2
=
(
k
+ 1)(
k
+2)
2
Thus,
k
+1
∈
S
and the statement is true for all
n
∈
N
.
2. Let
S
be the set of integers for which the statement is true.
Since 1
2
=1=
1
·
2
·
3
6
=
6
6
,
1
∈
S.
Assume that the statement is true for some positive integer
n
=
k
Then, for
n
=
k
1
2
+2
2
+
+
k
2
k
2
=[
1
2
2
+
+
k
2
k
2
=
k
(
k
+ 1)(2
k
6
k
2
=
k
(
k
+ 1)(2
k
+1)+6(
k
2
6
=
(
k
+ 1)(
k
+ 2)(2[
k
+1]+1)
6
Thus,
k
∈
S
and the statement is true for all
n
∈
N
.
3. Let
S
be the set of integers for which the statement is true.
Since 1 =
1

r
1

r
,
1
∈
S.
Assume that the statement is true for some positive integer
n
=
k
Then, for
n
=
k
1+
r
+
+
r
k
+
r
k
+1
1
+
r
+
+
r
k
]+
r
k
+1
=
1

r
k
+1
1

r
+
r
k
+1
=
1

r
k
+1
+(1

r
)
r
k
+1
1

r
=
1

r
k
+2
1

r
Thus,
k
∈
S
and the statement is true for all
n
∈
N
.
4. Let
S
be the set of integers for which the statement is true.
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This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.
 Spring '08
 Staff
 Integers

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