Let n be an integer greater than 1. Prove that n^2 - 2 - (n-2)^2 is an even number.
To show a number N is even, we must be able to express it in the form N = 2x for some other whole number x. Let's try to manipulate the given expression to see if we can put it in this form. By expanding the squared bracket and gathering like terms (be careful of the minus sign in front of the bracket!), we see that:
n2 - 2 - (n-2)2 = n2 - 2 -n2 +4n - 4 = 4n - 6
Now then, our new aim is to find x such that 4n - 6 = 2x. By dividing both sides of this equation by 2, we see that x = 2n-3. Since 2n - 3 will always be a whole number, we have shown that n2 - 2 - (n-2)2 = 2(2n-3), and so we are done as we have put the expression in the desired form.
