Discrete Algorithmic Mathematics, Third Edition
Written in a lively, clear style that talks to the reader, this book is unique for its emphasis on algorithmics and the inductive and recursive paradigms as central mathematical themes.
TABLE OF CONTENTS Discrete Algorithmic Mathematics, Third Edition
5.2: Modeling with Difference Equations
5.2 Modeling with Difference Equations
The purpose of this section is to take you through the process of setting up difference equations. We won t solve the equations we set up, but defer that until Section 5.3 or later. We have set up a few difference equations before, so in this section we ll emphasize examples that are harder or that illustrate applications not touched on earlier. However, it s a good idea to start with something simple and familiar.
Example 1
Describe arithmetic and geometric sequences using difference equations.
Solution Probably you have seen arithmetic sequences defined as sequences of the form
and geometric sequences as sequences of the form
An arithmetic sequence is one that starts at some value a and goes up by some fixed difference d each time. Well, this last sentence is just a difference equation in words. So s 0, s 1, . is an arithmetic sequence if and only if
Similarly, a geometric sequence is one that starts at some value a and is changed by a fixed multiplicative ratio r each time. So t 0, t 1, is a geometric sequence if and only if
Example 2
Describe arithmetic and geometric series using difference equations.
Solution A series is the sum of terms from a sequence. So an arithmetic series is
and a geometric series is
Even if we are only interested in these sums for one specific value of n
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