CSCE 235
Homework
Assignment 9
Assigned: April 7, 2008
Due: 12:30 p.m. April 11, 2008
(Homework 5
minutes late will not be accepted)
Recursively Defined Sequences
1. (5
points) Give recursive definitions of
each of the following sequences. For each,
give the values of the first two terms of the sequence as 
and 
; then define 
where k can
be 0, 1, or 2; and finally 
.
(a) 5, 3, 1, -1, -3, …
(b) 4, 1, 3, -2, 5, -7, 12, -19, 31, …
(c) 1, 2, 0, 3, -1, 4, -2, …
2. (10 points) Consider a sequence defined by 
, 
for 
. Prove by
induction that the sequence can be expressed as 
.
3. (15 points) A sequence is defined recursively by 
, 
, and 
for .
(a) Find the first five terms of this sequence.
(b)
Guess a formula for 
.
(c) Prove by induction that your guess in (b) is correct.
Recursion Relations
4. (70 points) In each of the following cases, give
an explicit formula for .
(a) 
, 
, and 
for .
(b) , 
for 
.
(c) 
, 
, and 
for .
(d) 
, 
, and 
for . Here c
and d are unspecified constants.
(e) , 
, and 
for .
(f) 
, 
, and 
for .
(g) , 
, and 
for .
Problems
based on (Goodaire and Parmenter
2002).