Question

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

Answer 1

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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. 

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Answer 2

$\huge\star\mathfrak\blue{{Answer:-}}$

Now (n+1)−(n−1)=2 which is not possible since any two perfect squares differ by at least 3. Hence there is no positive integer n for which is a rational number.