285
L26 Mathematical Induction; Binomial Theorem
Mathematical Induction
Mathematical induction
is a method for proving
mathematical statements which involve natural numbers.
We denote a statement
( )
A n
,
n
is natural.
Example
:
( )
A n
:
2
4
6
...
(2 )
(
1)
n
n n
+
+
+
+
=
+
,
1, 2, 3,
n
=
…
.
The Principle of Mathematical Induction:
Suppose that the following two conditions are
satisfied with regard to a statement
( )
A n
:
1.
( )
A n
is true for
0
n
n
=
.
2.
If
( )
A n
is true for
n
k
=
(
0
k
n
≥
), it is also
true for
1
n
k
=
+
.
Then
( )
A n
is true for all natural numbers
0
n
n
≥
.
286
Example
: Use mathematical induction to prove that the
formulas are valid.
( )
A n
:
(
1)
1
2
3
...
2
n n
n
+
+
+
+
+
=
,
1
n
≥

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