Show that the square of any odd number is an odd number
We can talk about any integer (or whole number) with the variable n. n could be any integer, 2, 3 or 100 or 5 billion and 1!If n is any integer, then what numbers could be written as 2n? These are the even numbers. How do I know they would all be even? (They can be divided by 2)How could I generalise all the odd numbers in the same way? (2n+1)We are interested in looking at the squares of odd numbers, so (2n+1)2 or (2n+1)(2n+1)= 4n2+ 4n + 1 We want to know if this is an odd number. We already know that an odd number can be written as an even number plus 1 and we can therefore rearrange: 4n2+ 4n + 1 = 2(2n2+ 2n) + 1, 2(2n2+ 2n) has to be even as whatever is in the brackets is then multiplied by 2. Therefore 2(2n2+ 2n) + 1 has to be odd for any integer n!
