$$\eqalign{ & {\text{If m}} = \sqrt {5 + \sqrt {5 + \sqrt {5\,......} } } \cr & {\text{n}} = \sqrt {5 - \sqrt {5 - \sqrt {5\,......} } } \cr} $$
then among the following the relation between m & n holds is.

Solution (By Examveda Team)

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