150
Solutions to Exercises
(d) The generating function for this geometric sequence is
1~""
as in PAUSE 10.
(e) The generating function is 3
+
3x
+
3x
2
+
3x
3
+ .
..
=
3(1
+
x
+
x
2
+ .
.. )
=
1~""
(f)
[BB]
The generating function is
1
+
x
2
+
x4
+
x
6
+ .
..
= 1
+
x
2
+
(x
2
)2
+
(x
2
)3
+
..
'.
This is
obtained by replacing
x
by
x
2
in (12). The generating function is
1':",2'
(g) The generating function is
1 
2x
+
3x
2

4x
3
+ .
.
'.
This is obtained by replacing
x
by
x
in (13). The generating function is
(1':",)2'
3.
[BB]
f(x)
=
ao
+
a1X
+
a2x2
+ .
.. +
anx
n
+
..
.
2xf(x)
=
2aox
+
2a1x2
+ .
.. +
2an_1Xn
+
..
.
Subtracting gives
f(x)

2xf(x)
=
ao
+
(a1

2ao)x
+ .
.. +
(an

2an_1)X
n
+ .
..
=
1 since
ao
=
1
and
an

2an1
=
0 for
n
~
1. Thus,
f(x)
_1_ = 1
+
2x
+
(2X)2
+ .
.. +
(2x)n
+ .
..
12x
1
+
2x
+
4x
2
+ .
.. +
2
n
x
n
+ .
..
We conclude that
an
=
2n.
Our solution is the same as the sequence defined in (5) of Section 5.2
except that it has one additional term,
ao
=
1.
4. We have
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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