College Algebra 9th txtbk

SECTION 8–2 Mathematical Induction 511
8-2 Mathematical Induction
Z Using Counterexamples
Z Using Mathematical Induction
Z Additional Examples of Mathematical Induction
Z Three Famous Problems
Many of the most important facts and formulas in this book have been stated as theorems.
But a theorem is not a theorem until it has been proved, and proving theorems is one of the
most challenging tasks in mathematics. There is a big difference between being pretty sure
that a statement is true, and proving that statement. Let’s look at an example.
Suppose that we are interested in the sum of the first n consecutive odd integers, where
n is a positive integer. We can begin by writing the sums for the first few values of n to see
if we can observe a pattern:
1 1
1 3 4
1 3 5 9
1 3 5 7 16
1 3 5 7 9 25
Is there any pattern to the sums 1, 4, 9, 16, and 25? You most likely noticed that each is a
perfect square and, in fact, each is the square of the number of terms in the sum. So the
following conjecture* seems reasonable:
CONJECTURE P: For each positive integer n,
1 3 5 . . . (2n 1) n 2
(Recall that the general term 2n 1 was used to list the odd positive integers in the last
section.)
At this point, you may be pretty sure that our conjecture is true. You might even look
at the previous five calculations and think that we have proved our conjecture. But in actuality,
all we have proved is that the conjecture is true for n 1, 2, 3, 4, and 5. We are trying
to prove that it is true for every positive integer, not just those five! With that in mind, continuing
to check the conjecture for specific n’s like 6, 7, 8, . . . is pointless: You can keep
trying for the rest of your life, but you will never be able to check every positive integer.
Instead, in this section, we will use a much more powerful tool called mathematical induction
to prove conjectures. Before we learn about this method of proof, we first consider how
to prove that a conjecture is false.
n 1
n 2
n 3
n 4
n 5
Table 1
n n 2 n 41
Prime?
Z Using Counterexamples
Consider the following conjecture:
1 41 Yes
2 43 Yes
3 47 Yes
4 53 Yes
5 61 Yes
CONJECTURE Q: For each positive integer n, the number n 2 n 41 is a prime number.
Since the conjecture states that this fact is true for every positive integer n, if we can
find even one positive integer n for which it is false, then the conjecture will be proved false.
A single case or example for which a conjecture fails is called a counterexample. We
checked the conjecture for a few particular cases in Table 1. From the table, it certainly appears
*A conjecture is a statement that is believed to be true, but has not been proved.