If $$m = \sqrt {5 + \sqrt {5 + \sqrt {5.....} } } $$      and $$n = \sqrt {5 - \sqrt {5 - \sqrt {5.....} } } $$      then among the following the relation between m & n holds is?

Solution (By Examveda Team)

$$\eqalign{ & {\text{Let }}m = \sqrt {5 + \sqrt {5 + \sqrt 5 } } \cr & {\text{Factor}} = \left( a \right) \times \left( {a + 1} \right) \cr & {\text{Here }}m = a + 1 \cr & \Rightarrow m - 1 = a\,.........(i) \cr & {\text{Let }}n = \sqrt {5 - \sqrt {5 - \sqrt 5 } } \cr & {\text{Factor}} = \left( a \right) \times \left( {a + 1} \right) \cr & {\text{Here }}n = a\,.........(ii) \cr & {\text{From (i) & (ii)}} \cr & \Leftrightarrow m - 1 = n \cr & \Leftrightarrow m - n - 1 = 0 \cr} $$