Question

show that there is no positive integer n for which root n-1+root n+1 is rational

i want step by step

Answer 1

hi!
we have √(n-1)+√(n+1)
let √(n-1)+√(n+1) is a rational then there exist co prime a and b such that

√(n-1)+√(n+1) = a/b -------(1)

=> 1/√(n-1)+√(n+1) = b/a [taking reciprocal on both sides] ----(2)

let us rationalize equation (2)
we get ,
=> [√(n+1)-√(n-1)]/2 = b/a

=> √(n+1)-√(n-1)= 2b/a ------(3)
adding (1) and (3)
we get
2√(n+1)= (b²+a²)/ab

=> √(n+1)= (b²+a²)/2ab
=> √(n+1) is rational [since a,b are integers]
=> n+1 is square of rational

subtracting (3) from (1) we get
2√(n-1)= a/b-2b/a

=> √(n-1)=(a²-2b²)/2ab
=>√(n-1 ) is a rational [ since a and b are integers ]
n-1 is also square of a rational number.
but two perfect square differ at least by 3
hence, there is no positive integer n for which √(n-1)+√(n+1) is rational .