CS173: Discrete Mathematical Structures
Fall 2005
Homework #6
Due 10/14/05
1. Give a recursive definition of the sequence
!
{an},n=1,2,3,K
if
a.
!
an=4n"2
.
b.
!
an=1+("1)n
.
c.
!
an=n(n+1)
.
2. Let S be the subset of the set of ordered pairs of integers defined recursively by:
Basis step:
!
(0,0) "S.
Recursive step: if
!
(a,b)"S
, then
!
(a+2,b+3) "S
and
!
(a+3,b+2) "S
a. List the elements of S produced by the first five applications of the
recursive definition (including the basis step).
b. Use induction to show that
!
5a+b
when
!
(a,b)"S
. (That is, the
remainder is zero when
!
a+b
is divided by 5.)
3. Suppose you have a function that accepts a music genre as input, and gives some
song from that genre as output, f: Genres
Songs. Denote dance music (a music
genre) by D, and rock (another music genre) by R. A playlist is just a list of
songs. For example, f(D)f(D) is a playlist of two dance songs. Give an inductive
definition of a good playlist, P, which is defined to be a playlist with more dance
songs than rock songs. (You should assume that no song belongs to more than one
genre.)
4. a. Devise a recursive algorithm to find the nth term of the sequence defined by
!
a0=1
,
!
a1=2
,
!
a2=3
, and
!
an=an"1+an"2+an"3
, for
!
n=3,4,5K
.
b. Devise an iterative algorithm to find the nth term of the sequence defined in
part a.
c. Is the algorithm from part a or the algorithm from part b more efficient?
Explain.