Question

show that there is no positive integer n
for which √n+1+√n+1 is rational.

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Answer 1

Explanation:

      ------ 1

q/p=1/\sqrt{n-1} + \sqrt{n+1}n−1+n+1  = \sqrt{n-1} + \sqrt{n+1}n−1+n+1 /-2

2q/p= \sqrt{n-1} - \sqrt{n+1}n−1−n+1                                                ------- 2

adding 1 and 2 

2\sqrt{n+1} = \frac{p}{q}+ \frac{2q}{p} = p^{2} + 2 \frac{ q^{2} }{pq}2n+1=qp+p2q=p2+2pqq2

\sqrt{n+1} [/tex] = p/q + 2q/p = px^{2}x2 + 2 q^{2}2q2 /2pq --3

subtracting 1 from 2

 

\sqrt{n-1}n−1 = \frac{p}{q} - 2 \frac{q}{p} = \frac{ p^{2} +2q^{2} }{2pq}qp−2pq=2pqp2+2q2  ------- 4

from 3 and 4