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 a real number if and only if

Solution :
$$\eqalign{ & {\left( {1 + i} \right)^{{n_1}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}} \cr & = {\left( {1 + i} \right)^{{n_1}}} + {\left( {1 - i} \right)^{{n_1}}} + {\left( {1 + i} \right)^{{n_2}}} + {\left( {1 - i} \right)^{{n_2}}} \cr & {\text{Using }}1 + i = \sqrt 2 \left( {\cos \frac{\pi }{4} + i\sin \frac{\pi }{4}} \right) \cr & {\text{and 1}} - i = \sqrt 2 \left( {\cos \frac{\pi }{4} - i\sin \frac{\pi }{4}} \right) \cr} $$
We get the given expression as
$$\eqalign{ & = {\left( {\sqrt 2 } \right)^{{n_1}}}\left[ {\cos \frac{{{n_1}\pi }}{4} + i\sin \frac{{{n_1}\pi }}{4}} \right] + {\left( {\sqrt 2 } \right)^{{n_2}}}\left[ {\cos \frac{{{n_2}\pi }}{4} + i\sin \frac{{{n_2}\pi }}{4}} \right] + {\left( {\sqrt 2 } \right)^{{n_2}}}\left[ {\cos \frac{{{n_2}\pi }}{4} - i\sin \frac{{{n_2}\pi }}{4}} \right] \cr & = {\left( {\sqrt 2 } \right)^{{n_1}}}\left[ {2\cos \frac{{{n_1}\pi }}{4}} \right] + {\left( {\sqrt 2 } \right)^{{n_2}}}\left[ {2\cos \frac{{{n_2}\pi }}{4}} \right] \cr} $$
= real number irrespective the values of $${{n_1}}$$ and $${{n_2}}$$
∴ $$(D)$$ is the most appropriate answer.