MATH 164, Spring 2001 Due Date: Name(s):
Honors Project 6: Finite Differences
Introduction
In this project we view a sequence {an} to be a function a : Z  R which associates to every integer
n  Z the real number an  R. Thus, for example, if an = n3
then some of the values of a are:
n : . . . -3 -2 -1 0 1 2 3 . . .
an = n3
: . . . -27 -8 -1 0 1 8 27 . . .
We define the forward difference operator + as the function which associates to each sequence a the
sequence +a whose nth term is
(+a)n = an+1 - an,
the backward difference operator - as the function which associates to each sequence a the sequence -a
whose nth term is
(-a)n = an - an-1,
and the sum operator  as the function which associates to each sequence a the sequence a whose nth term
is
(a)n =
n
i=0
ai.
Questions
1. If
an =
0 if n < 0
1
2n
if n  0 ,
find, in simplest form: a) (+a)n, b) (-a)n, and c) (a)n.
2. What general rules apply to computing +, -, and ? (For example, what can you say about +ca
where c is a constant and a is a sequence? What can you say about +(a  b) where a and b are
sequences? How about 1, n, n2
? Hint: Consider Sections 3.3 and 5.2 of the text.)
3. What:
a) is the composition   +? That is, what is the value of
((  +) a)n = ( (+a))n =
n
i=0
(+a)i =
n
i=0
(ai+1 - ai)
for any sequence a, in simplest form? How about   -?
b) is the composition +  ? That is, what is the value of
((+  ) a)n = (+ (a))n = (a)n+1 - (a)n =
n+1
i=0
ai -
n
i=0
ai
for any sequence a, in simplest form? How about -  ?
c) result (from MATH 163) do parts (a) and (b) remind us of?