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MATHEMATICAL INDUCTION
Original Notes adopted from September 11, 2001(Week 1)
c
±
P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics
The Natural Numbers
The set of
natural numbers
is the set
{
1
,
2
,
3
,
4
,
5
,...
}
,
which is denoted by
N
(or sometimes
N
.)
Principle of Mathematical Induction
If
S
⊂
N
such that
a) 1
∈
S
b) (
k
+ 1)
∈
S
whenever
k
∈
S
,
then
S
=
N
.
Example.
Prove: 1
2
+ 2
2
+ 3
2
+
...
+
n
2
=
n
(
n
+1)(2
n
+1)
6
.
Let
S
=
{
m
: 1
2
+ 2
2
+ 3
2
+
...
+
m
2
=
m
(
m
+1)(2
m
+1)
6
}
. We need to show that
S
=
N
.
By Mathematical Induction, it will suﬃce to show that
a) 1
∈
S
b) (
k
+ 1)
∈
S
whenever
k
∈
S
Is 1
∈
S
?
1
2
=
1(1+1)(2+1)
6
= 1
Yes.
Now suppose
k
∈
S
, that is,
1
2
+ 2
2
+ 3
2
+
...
+
k
2
=
k
(
k
+ 1)(2
k
+ 1)
6
Then we must show that
1
2
+ 2
2
+ 3
2
+
...
+
k
2
+ (
k
+ 1)
2
=
(
k
+ 1)(
k
+ 2)(2
k
+ 2 + 1)
6
First,
1
2
+ 2
2
+ 3
2
+
...
+
k
2
=
k
(
k
+ 1)(2
k
+ 1)
6
,
and so 1
2
+ 2
2
+ 3
2
+
...
+
k
2
+ (
k
+ 1)
2
=
k
(
k
+ 1)(2
k
+ 1)
6
+ (
k
+ 1)
2
=
k
(
k
+ 1)(2
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 Spring '10
 Applebaugh
 Math, Natural Numbers, Mathematical Induction

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