letsqrt(n-1)+sqrt(n+1)be a rational number which can be expressed as p/q, p and q are integers and coprime. q is not equal to 0squaring on both sides we get n-1+n+1+2sqrt(n^2-1)2n+2sqrt(n^2-1)=p^2/q^22(n+sqrt(n^2-1))=p^2/q^22(n+sqrt(n^2-1))q^2=p^2this mean 2 dividesp^2and also divides p.then let p=2k for any integer kthen 2(n+sqrt(n^2-1))=(2k)^2/q^22(n+sqrt(n^2-1))=4k^2/q^2q^2=2k^2/(n+sqrt(n^2-1))so 2 dividesq^2and also qp and q have common factors 2 which contradicts the fact that p and q are co-primes which is due to our wrong assumption. sosqrt(n-1)+sqrt(n+1)is irrational.

Understanding the Expression
To show that the expression √n + 1 + √n - 1 is not rational for any positive integer n, we start by simplifying it:
- The expression can be rewritten as √n + √n = 2√n.
Analyzing Rationality
Now, we need to consider when 2√n can be rational:
- A number is rational if it can be expressed as a fraction of two integers.
- For 2√n to be rational, √n itself must also be rational.
Conditions for Rationality of √n
- The square root of a number is rational if and only if that number is a perfect square.
- Therefore, n must be a perfect square for √n to be rational.
Exploring Perfect Squares
Let’s denote n as k², where k is a positive integer:
- Then, √n becomes k, and the expression simplifies to 2k.
Contradiction with Rationality
However, we need to analyze the original expression:
- The original expression is √n + 1 + √n - 1, which simplifies to 2√n.
- If n is not a perfect square, √n will be irrational, making 2√n irrational as well.
Conclusion
- Therefore, there is no positive integer n for which √n + 1 + √n - 1 is rational.
- In summary, if n is not a perfect square, the expression remains irrational, proving that no positive integer n satisfies the condition of the expression being rational.