Massachusetts Institute of Technology
6.042J/18.062J, Spring ’10
: Mathematics for Computer Science
April 26
Prof. Albert R. Meyer
revised April 26, 2010, 1303 minutes
Solutions
to
In-Class
Problems
Week
12,
Mon.
Problem
1.
The famous mathematician, Fibonacci, has decided to start a rabbit farm to ﬁll up his time while
he’s not making new sequences to torment future college students. Fibonacci starts his farm on
month zero (being a mathematician), and at the start of month one he receives his ﬁrst pair of
rabbits. Each pair of rabbits takes a month to mature, and after that breeds to produce one new
pair of rabbits each month. Fibonacci decides that in order never to run out of rabbits or money,
every time a batch of new rabbits is born, he’ll sell a number of newborn pairs equal to the total
number of pairs he had three months earlier. Fibonacci is convinced that this way he’ll never run
out of stock.
(a)
Deﬁne the number,
r
n
, of pairs of rabbits Fibonacci has in month
n
, using a recurrence rela-
tion. That is, deﬁne
r
n
in terms of various
r
i
where
i < n
.
Solution.
According to the description above,
r
0
= 0
and
r
1
= 1
. Since the rabbit pair received at
the ﬁrst month is too young to breed,
r
2
= 1
as well. After that,
r
n
is equal to the number,
r
n
−
1
,
of rabbit pairs in the previous month, plus the number of newborn pairs, minus the number,
r
n
−
3
,
he sells. The number of newborn pairs equals to the number of breeding pairs from the previous
month, which is precisely the total number,
r
n
−
2
, of pairs from two months before.
Thus,
r
n
=
r
n
−
1
+ (
r
n
−
2
−
r
n
−
3
)
.
(b)
Let
R
(
x
)
be the generating function for rabbit pairs,
R
(
x
)
::=
r
0
+
r
1
x
+
r
2
x
2
+
·
.
Express
R
(
x
)
as a quotient of polynomials.
Solution.
Reasoning as in the derivation of the generating function for the orginal Fibonacci num-
bers, we have
R
(
x
)
=
r
0
+
r
1
x
+
r
2
x
2
+
r
3
x
3
+
r
4
x
4
+
···
.
−
xR
(
x
)
=
−
r
0
x
−
r
1
x
2
2
−
r
2
x
3
3
−
r
3
x
4
4
− ···
.
−
x
2
R
(
x
)
=
−
r
0
x
−
r
1
x
−
r
2
x
− ···
.
x
3
R
(
x
)
=
+
r
0
x
3
+
r
1
x
4
+
···
.
R
(
x
)(1
−
x
−
x
2
+
x
3
)
=
r
0
+
(
r
1
−
r
0
)
x
+
(
r
2
−
r
1
−
r
0
)
x
2
+
0
x
3
+
0
x
4
+
···
.
=
0
+
1
x
+
0
x
2
.
Creative Commons
2010,
Prof. Albert R. Meyer
.