Question

plz solve this question.

Answer 1

HELLO DEAR,




let √(n−1) +√(n+1 )be a rational number which can be expressed as p/q, p and q are integers and coprime. q is not equal to 0
squaring on both sides we get n-1+n+1+2 

$\sqrt{ {n}^{2} - 1} \\ = > (2n + 2 \sqrt{ {n}^{2} - 1} ) = \frac{ {p}^{2} }{ {q}^{2} } \\ = > 2(n + 1 \sqrt{ {n}^{2} - 1} ) = \frac{ {p}^{2} }{ {q}^{2} } \\ = > 2(n + \sqrt{ {n}^{2} - 1} ) {q}^{2} = {p}^{2}$

THIS MEAN 2 Divides p2 AND Also DIVIDES p.
THEN Let P=2k FOR ANY INTEGER k

THEN,

$2(n + \sqrt{ {n}^{2} - 1 } ) = \frac{( {2k})^{2} }{ {q}^{2} } \\ = > 2(n + \sqrt{ {n}^{2} - 1} ) = 4 \times \frac{ {k}^{2} }{ {q}^{2} } \\ = > {q}^{2} (n + \sqrt{ {n}^{2} - 1} ) =2 {k}^{2} \\ = > {q}^{2} = \frac{2 {k}^{2} }{(n + \sqrt{ {n}^{2} - 1} )}$



 
SO 2 Divides q² AND Also q

p AND q HAVE COMMON FACTORS 2 WHICH CONTRADICTS THE FACT THAT p AND q ARE CO-PRIMES WHICH IS DUE TO OUR WRONG ASSUMPTION. So 
√(n−1)+√(n+1 ) IS IRRATIONAL.


I HOPE ITS HELP YOU DEAR,
THANKS