G.P 4

(aⁿ⁺¹+bⁿ⁺¹)/(aⁿ+bⁿ) = √ab

=((a/b)ⁿ⁺¹+1)/((a/b)ⁿ+1) = √a/b

=((a/b)ⁿ⁺¹+1)= ((a/b)ⁿ+1)x√a/b

=(a/b)ⁿ⁺¹+1= (a/b)^n+1/2+(a/b)^1/2

(a/b)ⁿ⁺¹ - (a/b)^n+1/2 = (a/b)^1/2-1

(a/b)^n+1/2 {(a/b)^1/2 -1} = (a/b)^1/2-1

{(a/b)^n+1/2-1} {(a/b)^1/2 -1} = 0

now did you get it?

only when n=-1/2

divide the numerator by bⁿ⁺¹ and the denominator by bⁿ

simultaneously divide the RHS by b