Question

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

Answer 1

p/q =                                                ------ 1

q/p=1/  = /-2

2q/p=                                                ------- 2

adding 1 and 2 



\sqrt{n+1} [/tex] = p/q + 2q/p = p +  /2pq --3

subtracting 1 from 2
 
 =   ------- 4

from 3 and 4

 and  are rational numbers as  and   

are rational for integer p and q .

here n+1 and n-1 are perfect square of positive integer.

now,(n+1)-(n-1)=2 which is not possible since any perfect square differ by
 
at least 3.

thus there is no positive integer n which  and  is rational.