prove that for any positive integer and √2n+1  + √2n+3 is not rational;

Asked by Sanmit Ratnaparkhi | 1st May, 2014, 08:51: PM

Expert Answer:

Lets assume that the given number is rational, which can be expressed as p/q , where p and q are intergers
 
?2n+1 +?2n+3 = p/q
 
squaring the both sides, we get
 
2n+1 +2n+3 +2(?2n+1) (?2n+3)= p2/q2
 
4n+4 +2(?2n+1) (?2n+3)= p2/q2
 
2(?2n+1) (?2n+3) = 4n+4 - p2/q2
 
irrational*irrational= rational - rational
 
irrational = rational, which can not be true.
 
Therefore, the given number is irrational.

Answered by Avinash Soni | 4th May, 2014, 03:50: PM

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