Question

Show that the square of an oddnumber is an odd number using a direct proof.

Answer 1

Consider that for any integer (n), (2n+1) is an odd number.

By expansion and collecting terms, (2n+1)^2 = ______.

Since _______ is an integer it follows that (2n+1)^2 is odd.

Therefore the square of any odd number is also an odd number.

Answer 2

any odd number can be written in the form 2n + 1 , where n is a positive integer.

(2n+1)^2 = 4n^2 + 4n + 1
= 4(n^2 + 1) + 1

since any integer multiplied by 4 is even, the term 4(n^2 + 1) is even , adding 1 to it would make it odd.