A proof is a
reasoned argument to establish the truth of a mathematical result.
For example the sequence
12 = 1
32 = 9 = 8 + 1
52 = 25 = 24 + 1
72 = 49 = 48 + 1
92 = 81 = 80 + 1
suggests that the square of an odd number is always 1 more than
a multiple of 8. But is it always true ? A proof follows:
Any odd number
can be expressed as 2n + 1 where n is a whole number.
(2n + 1)2 = 4n2+ 4n+1
= 4(n2+ n) + 1
= 4n (n + 1) + 1
Now n and n+1 are consecutive numbers so one of them is even and
has a factor of 2. Hence n(n + 1) = 2m for some whole number m, and
it follows that (2n + 1)2 = 8m + 1. To prove a result is to construct
a proof for the result.
