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
+ 1)
=
k
(
k
+ 1) + 2(
k
+ 1)
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,
1
∈
S.
Assume that the statement is true for some positive integer
n
=
k
Then, for
n
=
k
+ 1,
1
2
+ 2
2
+
· · ·
+
k
2
+ (
k
+ 1)
2
=
[1
2
+ 2
2
+
· · ·
+
k
2
] + (
k
+ 1)
2
=
k
(
k
+ 1)(2
k
+ 1)
6
+ (
k
+ 1)
2
=
k
(
k
+ 1)(2
k
+ 1) + 6(
k
+ 1)
2
6
=
(
k
+ 1)(
k
+ 2)(2[
k
+ 1] + 1)
6
Thus,
k
+ 1
∈
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,
1
∈
S.
Assume that the statement is true for some positive integer
n
=
k
Then, for
n
=
k
+ 1,
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
+ 1
∈
S
and the statement is true for all
n
∈
N
.
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 Spring '08
 Staff
 Integers, Rational number, positive integer

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