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Section 11.4 Mathematical Induction
Mathematical induction is a method that is useful for proving mathematical formulas that might
otherwise be diﬃcult (or impossible) to prove.
Example.
Show that 1 + 2 + 3 +
···
+
n
=
n
(
n
+1)
2
for all
n
.
We note the following:
n
=1:1=
1(2)
2
=1
n
=2:1+2=
2(3)
2
=3
n
=3: 1+2+3=
3(4)
2
=6
···
We see that it is impossible to directly check all (inﬁnitely many) integers
n
to prove the validity
oftheformula1+2+3+
···
+
n
=
n
(
n
+1)
2
for all
n
. Thus, some method of proof, other than a
direct check, will be needed to establish the formulas’ validity for all
n
. One method of proof is
Mathematical Induction.
Toprovethat1+2+3+
···
+
n
=
n
(
n
+1)
2
by Mathematical Induction, we show that the formula
holds when
n
= 1, and then we show that whenever the formula holds for some integer
n
, then the
formula also holds for the next larger integer
n
+ 1. These two facts imply that the formula holds
for all
n
. The details of the proof will be shown below.
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This note was uploaded on 12/11/2011 for the course MAC 1140 taught by Professor Kutter during the Fall '11 term at FSU.
 Fall '11
 Kutter
 Algebra, Formulas, Mathematical Induction

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