5.4 PROOF BY MATHEMATICAL INDUCTION 1mGeneralization. Many theorems or axioms whose familiar statement says
something about two objects are generalized to "any finite number" by an
induction proof.
EXAMPLE 6 Use induction and the distributive axiom to prove the law of
generalized distributivity of multiplication over addition, that is, for any
positive integer n,
a(b, + b, +... + b,) = ab, + ab2 +. + ab,,
where a, b,,... , b, are real numbers.
Solution In proofs of this type induction is done on the number m of objects
involved, in this case the number of real numbers over which we are dis-
tributing a. Also, the "known" distributive law is simply the case m = 2,
which is thereby known to be true. Let S be the set of those positive
integers m such that "distributivity across m real numbers" is valid.
Condition (i) of the induction principle says that ab, = ab,, which is
true. For condition (ii), assume m E S. This means that a(b, + b, +.
- b,) = ab, + ab2 +. + ab, for any real number a and for any m real
numbers b,, b,,... , b,. To show m + 1 E S, let a E R and let c,, c,,... ,
c,, em+, be any (m + 1) real numbers. Then
C, + ..- + c, + em+,)
a[(cl + c2 + + c,) + c,,,+~]
+ c2 + ..- + c,) + ac,+,
(ac, + ac, + + ac,,,) + ac,,,
ac, + ac, + + ac, + ac,, ,, as desired
Note, finally, that generalized distributivity can be expressed by sum-
mation notation, namely, the equation C; =, (ab,) = az =, b,. 0Notice that the step leading from line 2 to line 3 was based on the ordi-
nary distributive law (i.e., the case n = 2 of generalized distributivity),
whereas we went from line 3 to line 4 by means of the induction hypothesis.
The pattern of an essentially two-step argument, using first the known case
n = 2, followed by using the induction hypothesis, is the usual one when
induction is used for the purpose of generalization. Note that (d) of Exam-
ple 1 is a problem in this category, as are the two parts of Exercise 6.
It may be appropriate, at this point, to note that the results of Examples
5 and 6 are among a number of "basic properties of summation notation"
that are proved by induction. These include additional results such as
z,, c = nc (for any n E N and c E R) and C,, x, = z+;+i x,-, (for any
positive integers n and i, and real numbers x,, x,,... , x,). Generalizations
of these and other basic summation notation properties are the subject of
Exercise 15.