show that there is no positive integer n for which

Consecutive perfect squares starting from 1 are 1,4,9,16, 25 and so on.

For n-1 to be  a perfect square, n=2,5,10,17 ,26 and so on

For n+1 to be a perfect square, n=0,3,8,10,24 and so on

Clearly, both (n-1),(n+1) cannot be rational for any positive integral values of n .So

there is no positive integer n for which (n-1)+(n+1) is rational.