MATH 204. PROOF BY INDUCTION.
MRINAL RAGHUPATHI
These notes are an informal introduction to proof by induction.
A more careful presentation of
these ideas may be given during the course.
To motivate this technique we begin with a simple problem. What is the sum of the first
n
natural
numbers? What we are looking for is a formula for the sum 1 + 2 +
· · ·
+
n
. This number arises for
instance when you want to count the number of handshakes that are required at a business meeting
where there are
n
people, or the number of “clinks” at a toast.
Here is an elegant solution due to Gauss. Write the numbers from 1 to
n
in reverse and add:
1
2
· · ·
n
+
n
(
n

1)
· · ·
1
=
n
+ 1
n
+ 1
· · ·
n
+ 1
This calculation shows that 2(1 + 2 +
· · ·
+
n
) =
n
(
n
+ 1) from which we get
1 + 2 +
· · ·
+
n
=
n
(
n
+ 1)
2
.
We may not be as clever as Gauss. A good starting point is to compute the sum for some small
values of
n
and look for a pattern. Here are the first ten values:
(1)
1
,
3
,
6
,
10
,
15
,
121
,
28
,
36
,
45
,
55
.
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 Fall '08
 STAPLES
 Math, Linear Algebra, Algebra, Natural Numbers

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