show that there is no positive integer n so that root(n-1)+root(n+1) is rational

Suppose there exists a positive integer *n* for which is a rational number.

, where *p* and *q* positive integers and *q* ≠ 0.

From (1) and (2),

⇒ *n* + 1 and *n* – 1 perfect square of positive integers.

Now, (*n* + 1) – (*n* – 1) = 2, which is not possible since **any two perfect squares differ by atleast 3.**

Hence, there is no positive integer *n* for which is a rational number.

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