prove that there is no positive integer n for which root n - 1 + root n + 1 is rational.

Suppose \(\displaystyle a=\sqrt{n-1}+\sqrt{n+1}\) is rational.

Then

\(\displaystyle a^2=2n+2\sqrt{(n+1)(n-1)}\)

\(\displaystyle \frac{a^2-2n}2=\sqrt{(n+1)(n-1)}\)

Hence \(\displaystyle \sqrt{(n+1)(n-1)}\) is rational, which implies that \(\displaystyle (n+1)(n-1)=n^2-1\) is a perfect square.

We get \(\displaystyle n^2-1=m^2\) for some integer m.

But the only solutions in integere are \(\displaystyle n=\pm1\) and \(\displaystyle m=0\).

But for n=1 we have \(\displaystyle a=\sqrt2\), which is irrational.

EDIT: Corrected typo n(n-1) to (n+1)(n-1) on one place.