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52 The Nuts and Bolts of Proof, Third Edition
Using the associative property of addition, we can write:
1 + 3 + 5 + ... + (2n - 1) + [2{n + 1) - 1]
= {1 + 3 + 5 + ... + (2n - 1)} + (2n + 1).
Then we can use the inductive hypothesis to obtain:
1 + 3 + 5 + ... + (2n - 1) + [2(n + 1) - 1]
= n^ + (2n+l)
= (n+l)l
Therefore, the formula given in the statement holds true for the sum of an
arbitrary number of odd natural numbers, by the principle of mathematical
induction. •
The principle of mathematical induction stated before Example 1 is also
known as the weak (or first) principle of mathematical induction, in contrast
to the strong (or second) principle of mathematical induction, usually stated
as follows:
Let P{n) represent a statement relative to a positive integer n. If:
1. P(t) is true, where t is the smallest integer for which the statement can
be made,
2. whenever P(k) is true for all numbers k with k = t,t-\-l,.. .,n,it follows
that P{n + 1) is true as well,
then P{n) is true for all n>t.
Let us see how to use this principle in a proof.
EXAMPLE 3. If n > 1 is a counting number, then either n is a prime
number or it is a product of primes.
Proof:
1. Base case. The statement is true for the smallest number we can
consider, which is 2. This number is indeed prime.
2. Inductive hypothesis. Assume the statement is true for all the numbers
between 2 and an arbitrary number k, including k.
3. Deductive proof Is the statement true for /c+ 1?
If /c + 1 is a prime number, then the statement is trivially true.
If /c +1 is not a prime number, then by definition it has a positive
divisor d such that d/ 1 and d> k+1.
Thus, k + 1 = dm with m / 1, m > /c + 1, and m > 2.