124
Consider a
(k
+
I)gon. Let
P
be a vertex and
Q,
R
the vertices
adjacent to
P.
Join
QR
as shown on the right. Since
k
?:
3,
the line
Q R
separates the figure into two distinct pieces, triangle
PQ
R
and
the
kgon
fonned
by
QR
and all sides
of
the original
(k
+
I)gon
except
PQ
and
P R.
By the induction hypothesis, the sum
of
the
interior angles of this new
kgon
equals
(k

2)
180
0
•
Also, triangle
PQR
has sum
of
interior angles equal to 180
0
•
Solutions to Exercises
The sum
of
the angles
of
the
(k
+
1 )gon is the sum
of
the angles in the
kgon
and in the triangle, that
is,
(k

2) 180
0
+
180
0
=
(k

1) 180
0
,
as
desired. By the Principle
of
Mathematical Induction, the
result is true for all
n
?:
3.
29.
We
prove this by induction on
n,
the number of straight lines drawn.
If
n
=
1,
we have two regions
which,
if
colored with different colors, gives a proper coloring using just two colors. (By "proper," we
mean that bordering countries have different colors.) Now suppose that
k
?:
1 and the statement is
true for
n
=
k;
that is, suppose that a map made by drawing
k
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 Summer '10
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 Graph Theory, Mathematical Induction, Inductive Reasoning, induction hypothesis, light coin

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