Advanced High-School Mathematics

SECTION 2.1 Elementary Number Theory 95

note, however that the difference equations leading to arithmetic se-
quences (un+1−un= d, n = 0, 1 , 2 ,...) are not homogeneous. We’ll
treat generalizations of the arithmetic sequences in Section 2.1.9, below.

We shall now separate the homogeneous and inhomogeneous cases:^24

Homogeneous difference equations

We shall consider a few commonly-occuring cases.

Linear. Given the monic polynomial C(x) we are trying to solve

C(u) = 0 for the unknown sequence u = (u 0 ,u 1 ,u 2 ,...). As-

sume that the polynomial is linear: C(x) =x−k, for some real

constantk; thus the difference equation assumes the form

un+1 = kun, n= 0, 1 , 2 ,... (2.2)

This says that each successive term isktimes the preceding term;

this is the definition of a geometric sequence with ratio k.

Clearly, then the solution is

un = knA, n= 0, 1 , 2 ,... (2.3)

whereAis an arbitrary constant. The solution given in equation

(2.3) above is called thegeneral solution of the first-order dif-

ference equation (2.2). Theparticular solutionis then obtained

by specifying a particular value forA.

(^24) The reader having studied some linear differential equations will note an obvious parallel!