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September 6, 2011
An inductive proof
•
Proposition: The
n
’th Fibonacci number is the number of
strings over
{
a,b
}
of length
n

2
with no consecutive
b
’s.
•
First step: make sure you know what everything means
•
What are the Fibonacci numbers?
•
Where does the indexing start?
•
How does the statement of the proposition need to be
cleaned up to be precise?
•
Is the proposition true for the smallest value of
n
?
•
If it is true for small values of
n
, say for all
n
≤
k
, why would it
be true for
n
=
k
+ 1
?
1
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September 6, 2011
So here is what we want to prove:
Proposition:
For every
n
≥
2
,
F
n
is the number of strings over
{
a,b
}
of length
n

2
with no consecutive
b
’s.
•
Proof:
Base case:
Prove it for
n
= 2
and
n
= 3
, by a hand check.
Induction step:
•
For any
n
, let
X
n
be the set of strings of length
n
with no
consecutive
b
’s. We have proven that

X
n

=
F
n
+2
for
n
= 0
and
n
= 1
. Now let
k
be any number
≥
1
and assume we
have proven that

X
n

=
F
n
+2
for all
n
≤
k
. We want to
conclude that

X
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 Fall '09

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