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140
Solutions to Exercises
any sequence adding to
n
or to
n
 1 can be turned into a sequence adding to
n
+ 1 by adding
a 1 or a 2 (respectively) at the end. Thus,
bn+1
=
bn
+
bn I.
Adding the initial conditions
b
l
=
1,
b
2
=
2 gives the desired recursive definition and proves that the sequence is the Fibonacci
sequence, starting with the second tenn.
55. (a) [BB] Each line of
n
 1 people determines two lines of
n
peoplethe first person in the new line
can be a man or a woman. So
an
=
2an
l.
The sequence is the sequence 2, 4, 8,
... of powers
of2.
(b) A line starts with a male or a female.
If
it starts with a female, there are
anI
ways to complete
the line. On the other hand, if it starts with a male, then the second person must be female. There
are
an2
ways then to complete such a line. Thus,
an
=
anI
+
an2.
Since
al
=
2 (M or F)
and
a2
=
3 (FF, MF, or PM), the next few tenns are 5, 8, 13, .
... This is that part of the Fibonacci
sequence from 2 onward.
56. (a) We use induction on
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 Summer '10
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 Graph Theory

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