Homework 3 Solutions
Note: These solutions are not necessarily model answers. Rather, they are designed to be tutorial in nature,
and sometimes contain a little more explanation than an ideal solution. Also, bear in mind that there may be
more than one correct solution.
1. A formula for the Fibonacci Numbers
Base cases
: For
n
= 0
the formula evaluates to zero, which is the correct value of
F
(0)
.
check
.
For
n
= 1
, the value of the formula is
φ

(1

φ
)
√
5
=
2
φ

1
√
5
=
2
1+
√
5
2

1
√
5
=
2 + 2
√
5

2
2
√
5
= 1 =
F
(1)
.
check
So both base cases are verified.
Inductive hypothesis
: Assume that the formula is valid for
0
≤
k
≤
n
, where
n
≥
1
is arbitrary. (Note that
we are using
strong
induction.)
Induction step
: We want to show that the formula is valid for
n
+ 1
. To do this, we first notice two facts that
follow from the definition of
φ
and that will be useful to us in the algebra below:
(a)
1 +
φ
=
φ
2
;
(b)
(1

φ
)
2
= 1

2
φ
+
φ
2
= 2

φ.
Now, using the recursive definition of the Fibonacci sequence we have
F
(
n
+ 1)
=
F
(
n
) +
F
(
n

1)
=
φ
n

(1

φ
)
n
√
5
+
φ
n

1

(1

φ
)
n

1
√
5
[by the inductive hypothesis]
=
1
√
5
bracketleftbig
φ
n

1
(
φ
+ 1)

(1

φ
)
n

1
(1

φ
+ 1)
bracketrightbig
=
1
√
5
bracketleftbig
φ
n

1
(
φ
+ 1)

(1

φ
)
n

1
(2

φ
)
bracketrightbig
=
1
√
5
bracketleftbig
φ
n

1
φ
2

(1

φ
)
n

1
(1

φ
)
2
bracketrightbig
[using (a) and (b) above]
=
1
√
5
bracketleftbig
φ
n
+1

(1

φ
)
n
+1
bracketrightbig
.
This completes the induction step, and thus we can conclude that the formula is valid for all
n
.
2. Strengthening the Induction Hypothesis
First note that
parenleftbigg
a
b
c
d
parenrightbiggparenleftbigg
1
0
1
1
parenrightbigg
=
parenleftbigg
a
+
b
b
c
+
d
d
parenrightbigg
,
so if we just try to use the inductive hypothesis that all the entries of the
n
th power
parenleftbigg
1
0
1
1
parenrightbigg
n
=
parenleftbigg
a
b
c
d
parenrightbigg
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 HAULLGREN
 Computer Science, Mathematical Induction, Recursion, Inductive Reasoning, Natural number, N×N

Click to edit the document details