a, b are positive reals such that 1/a+1/b = 1/ (a+b). If (a/b)^3+(b/a)^3 =2$\sqrt{n}$, where n is a natural number. Find the value of n.

Solution:

Given, 

$\frac{1}{a} +\frac{1}{b} =\frac{1}{a+b}$

$\frac{a+b}{ab}  =\frac{1}{a+b}$

$\frac{(a+b)^2}{a} =\frac{a+b}{a+b}$

$\frac{a^2+b^2+2ab}{ab}  =1$

$a^2+b^2+2ab  = ab$

$a^2+b^2 = ab - 2ab$

$a^2+b^2 = -ab$

as $a^2+b^2 > 0$  as a ,b are positive reals

So, positive real numbers will have positive squares so this can not be possible.

hence , $n$ would have No Real Positive Values.