文档介绍：¢¡£¡¥¤§¦¨¤§©
¤
 £© Section
 ¢¡£¡¥¤§¦¨¤§©  £
 £© Difference Equations
At this point almost all of our sequences have had explicit formulas for their terms. That
is, we have looked mainly at sequences for which we could write the nth term as an = f(n)
for some known function f. For example, if
n + 1
a = ,
n n2 + 3
11 101
then it is an easy matter pute explicitly, say, a10 = 103 or a100 = 10003 . In such
cases we are able pute any given term in the sequence without reference to any
other terms in the sequence. However, it is often the case in applications that we do not
begin with an explicit formula for the terms of a sequence; rather, we may know only
some relationship between the various terms. An equation which expresses a value of a
sequence as a function of the other terms in the sequence is called a difference equation.
In particular, an equation which expresses the value an of a sequence {an} as a function of
the term an−1 is called a first-order difference equation. If we can find a function f such
that an = f(n), n = 1, 2, 3, . . ., then we will have solved the difference equation. In this
section we will consider a class of difference equations that are solvable in this sense; in
the next section we will discuss an example where an explicit solution is not possible.
Example Suppose a certain population of owls is growing at the rate of 2% per year. If
we let x0 represent the size of the initial population of owls and xn the number of owls n
years later, then
xn+1 = xn + = ()
for n = 0, 1, 2, . . .. That is, the number of owls in any given year is equal to the number
of owls in the previous year plus 2% of the number of owls in the previous year. Equation
() is an example of a first-order difference equation; it relates the number of owls in
a given year with the number of owls in the previous year. Hence we know the value of a
specific xn once we know the value of x