epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Special Kinds of Theorems 51
use the equality stated in the inductive hypothesis:
perform algebraic steps:
n(n + 1)
• + (n+l)
2
n(n -f 1) + 2(n + 1)
2
(n+l)(n + 2)
Thus,
1 + 2 + 3H hn + (n+l) = r
V ^ ^ 2
(«+l) numbers
Therefore, the formula given in the statement holds true for all natural
numbers A; > 1, by the principle of mathematical induction. •
A different proof of the result stated in Example 1 can be found in the
previous chapter (see Example 3). This is one of the nice situations in which
several proofs of the same statement can be constructed using different
mathematical tools.
EXAMPLE 2. The sum of the first k odd numbers is equal to k^; that is:
Proof:
1 + 3 + 5 -h ... + (2fc - 1) := fc2
1. Base case. Does the equality hold true for /c= 1, the smallest number
that can be used? In this case, we are considering only one odd
number. Therefore, we have:
1 = 1^
Thus, the equahty is true for /c= 1.
2. Inductive hypothesis. Assume the equahty holds true for an arbitrary
collection of n odd numbers. Thus,
1 + 3 + 5 + ... + (2M - 1) = nl
3. Deductive proof. We want to prove that the equahty is true for n +1
odd numbers. Therefore, we need to check the equahty:
1 + 3 + 5 + ... -f (2/2 - 1) + [2{n 4-1) - 1] = (w + if.