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

Introduction and Basic Terminology 15
If we use the formula given above, we obtain 5(5 + l)/2 = 30/2 = 15.
Clearly, the two answers coincide, but this might be true just by chance. To
construct a proof, we need to find a mathematical relationship between the
sum of counting numbers and the formula given in the conclusion.
Proof: Let Sn denote the sum of the first n counting numbers; that is:
S„ = 1 + 2 + 3 + .. • + (n - 1) + n.
Because addition is a commutative operation, we can try rearranging the
numbers to write:
5„3zn + (n-l) + --. + 3 + 2+l.
Compare these two ways of writing S^:
5„ = 1 + 2 + • • • + (n - 1) + n
5„ = n + (n-l) + --- + 2+l.
If we add these two equations, we obtain:
or
or
25„ - (1 + n) + [2 + (n - 1)] + ... + [{n - 1) + 2] + (n + 1)
2^, = (n + 1) + (n + 1) + • •. + (n + 1) + (n + 1)
From this last equation, we obtain:
2Sn = n(n + 1).
n(n + 1)
Sn=-
2 '
n(n + 1)
Therefore, it is true that 1 + 2 + 3 + .. . + n
z
The section on Mathematical Induction includes a different proof of the
result presented in Example 3. The proof shown in Example 3 is known as
Gauss' proof.
EXAMPLE 4. If a and b are two positive integers, with a> b, then we can
find two integers q and r such that:
where 0<r <b and 0<q.
a = qb-{-r