9514 1404 393
Explanation:
We will prove by contradiction. We assume that the given sum is rational, and the ratio can be expressed in reduced form by p/q, where p and q have no common factors.
[tex]\sqrt{n-1}+\sqrt{n+1}=\dfrac{p}{q}\qquad\text{given}\\\\(n-1)+2\sqrt{(n-1)(n+1)}+(n+1)=\dfrac{p^2}{q^2}\quad\text{square both sides}\\\\2(n+\sqrt{n^2-1})=\dfrac{p^2}{q^2}\qquad\text{simplify}[/tex]
We note that this last equation can have no integer solutions (n, p, q) for a couple of reasons:
- for any integer n > 1, the root √(n²-1) is irrational (n² is a perfect square; n²-1 cannot be.)
- p²/q² cannot have a factor of 2, as √2 is irrational
There can be no integer n for which the given expression is rational.
