A proof is a
reasoned argument to establish the truth of a mathematical result.

For example the sequence

1^{2} = 1

3^{2} = 9 = 8 + 1

5^{2} = 25 = 24 + 1

7^{2} = 49 = 48 + 1

9^{2} = 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} = 4n^{2}+ 4n+1

= 4(n^{2}+ 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.