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

50 The Nuts and Bolts of Proof, Third Edition
EXAMPLE 1. Prove by induction that the sum of the first k natural
numbers is equal to k{k + l)/2.
Proof: We want to prove that the equahty:
l + 2 + 3 + ... + (fc-l) + fc = ^^^
^ ^ ' 2
k numbers
holds true for all /c > 1 natural numbers.
1. Base case. Does the equality hold true for /c= 1, the smallest number
that can be used?
^ ^ 1(1 + 1)
2
Thus, by using the given formula we obtain a true statement. This
means that the formula works for /c= 1.
2. Inductive hypothesis. Assume the formula works when we add the first
n numbers (fc = n). Thus,
l+2 + 3 + -- + (n-l) + n:::::^^ \
n numbers
3. Deductive proof. We want to prove that the formula holds true for the
next number, n+ 1. Thus, we have to prove that:
. . . r .. (n + l)[(n + 1) + 1]
l + 2 + 3 + -j + yz + (n+l)^- ^-^
(n+l) numbers
or, equivalently,
(n+l) numbers
To reach this goal, we will need to use the equaUty stated in the
inductive hypothesis:
l + 2 + 3 + --- + w + (nH-l)
associative property of addition of numbers:
== [1 + 2 + 3 + •. • + n] + (n + 1)