Answer:
Starting with the expression n² - 2 - (n - 2)², we can simplify it by expanding the second term using the formula for the square of a binomial:
n² - 2 - (n - 2)² = n² - 2 - (n² - 4n + 4)
Next, we can combine like terms:
n² - 2 - (n² - 4n + 4) = n² - n² + 4n - 6
Simplifying further, we get:
n² - 2 - (n - 2)² = 4n - 6
To prove that this expression is always even, we can show that it can be written in the form 2k, where k is an integer.
So, let's rewrite the expression as:
4n - 6 = 2(2n - 3)
Since 2n - 3 is an integer (because n is an integer greater than 1), we have shown that 4n - 6 can be written in the form 2k, where k is an integer.
Therefore, we have proved algebraically that n² - 2 - (n - 2)² is always an even number.