JEE Advance - Mathematics (1996 - No. 13)
For positive integers $${n_1},\,{n_2}$$ the value of the expression $${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}},$$
where $$i = \sqrt { - 1} $$ is real number if and only if
where $$i = \sqrt { - 1} $$ is real number if and only if
$${n_1} = {n_2} + 1$$
$${n_1} = {n_2} - 1$$
$${n_1} = {n_2}$$
$${n_1} > 0,\,{n_2} > 0$$