prove that there is no positive integer n for which root n - 1 + root n + 1 is rational.
is rational if and only is rational.
meaning that is rational and therefore (n-1)n is a perfect square.
Now since n-1 and n are coprime, both n-1 and n are perfect squares. So again you have two squares which differ by 1 and you can continue int he same way as I did in my first post. You get that the only solution is n=1.
BTW do you know the result, that a square root of an integer is rational if and only if this integer is a perfect square? If you did not learn this, you were probably supposed to look for a completely different solution. (Perhaps there is a simpler one and I have overlooked it...)