Question

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

Answer 1

Let us assume that there is a positive integer n for which √n-1 +√n+1 is rational.
and equal to p/q,
where p and q are positive integer (q is not equal to 0).
√n-1 +√n+1 = p/q eq.(1)
q/p =1/√n-1 +√n+1
=√n-1 +√n+1/(√n-1 +√n+1)(√n-1 -√n+1)
=√n-1 -√n+1/(n-1)-(n+1)
=√n-1 -√n+1/-2
=√n+1 -√n-1 =2q/p eq.(2)
apply (1) And (2),we get
2√n+1=p/q +2q/p =p²+2q²/pq
√n+1=p²+2q²/2pq eq.(3)
apply (1) - (2),
√n-1 = p²-2q²/2pq eq.(4)
from (3) and (4),we can say √n+1 and √n-1 both are rational because p and q both are rational . But it is possible only when (n+1) and (n-1) both are perfect square. But they differ by 2 and two perfect square never differ by 2 .so both (n+1) and (n-1) cannot be perfect square,hence there is no positive integer n for which √n-1 + √n+1 is rational.

Answer 2

Answer:

Step-by-step explanation: