Question

Show that thier is no positive integer n,
for which
$\sqrt{n - 1} + \sqrt{n - 1}$
is rational.​

Answer 1

Step-by-step explanation:

let us suppose that there exists a positive integer n for which

$\sqrt{n - 1} + \sqrt{n + 1}$

is rational . Then , there exist 2 co prime integers a and b such that

a/b =

$\sqrt{n - 1} + \sqrt{n + 1}$

let this be( i) where a/b > 0

a/b = root n-1 +root n+1 × rootn-1- root n+1/root n-1 - root n+1

( rationalizing the numerator )

a/b = (n-1) - (n+1)/ root n-1 -root n+1

a/b= -2 /root n-1 - root n+1

root n-1 - root n+1 = -2b/a..........(ii)

adding (i) and (ii)

a/b - 2b/a = 2×root n-1

1/2(a/b-2b/a) = root n-1 .........(iii)

subtracting (ii) from (i)

a/b + 2b/a = 2×root(n+1

1/2 (a/b + 2b/a) = root n+1 ........(iv)

root n+1 and root n-1 are rationals ( a and b are integers)

n+1 and n-1 are perfect squares of positive integers

this is not possible as any two perfect squares differ at least by 3

hence our assumption is wrong

there exists no positive integer for n

mark me the brainliest