Let
#sqrt(n+1)+sqrt(n-1)" is rational and can be expressed by " p/q #
where p and q prime to each other and#" "q!=0#
So
#sqrt(n+1)+sqrt(n-1)=p/q.........(1)#
Inverting (1) we get
#1/(sqrt(n+1)+sqrt(n-1))=q/p#
#=>(sqrt(n+1)-sqrt(n-1))/((sqrt(n+1)+sqrt(n-1))(sqrt(n+1)-sqrt(n-1)))=q/p#
#=>(sqrt(n+1)-sqrt(n-1))/2=q/p#
#=>(sqrt(n+1)-sqrt(n-1))=(2q)/p.....(2)#
Adding (1) and (2) we get
#2sqrt(n+1)=p/q+(2q)/p#
#=>sqrt(n+1)=(p^2+2q^2)/(2pq).....(3)#
Similarly subtracting (2) from (1) we get
#=>sqrt(n-1)=(p^2-2q^2)/(2pq).....(4)#
Since p and q are integers then eqution (3) and equation(4) reveal
that both#" "sqrt(n+1) and sqrt(n-1)#
#color (blue)(" are rational as their RHS rational")#
So both #(n+1) and (n-1) " will be perfect square"#
Their difference becomes #(n+1)-(n-1)=2#
But we know any two perfect square differ by at least by 3
Hence it can be inferred that there is no positive integer for which
#sqrt(n+1)+sqrt(n-1)" is rational"#
