• # question_answer For positive integers ${{n}_{1}},{{n}_{2}}$the value of the expression  ${{(1+i)}^{{{n}_{1}}}}+{{(1+{{i}^{3}})}^{{{n}_{1}}}}+{{(1+{{i}^{5}})}^{{{n}_{2}}}}+{{(1+{{i}^{7}})}^{{{n}_{2}}}}$where $i=\sqrt{-1}$  is a real number if and only if [IIT 1996] A) ${{n}_{1}}={{n}_{2}}+1$ B) ${{n}_{1}}={{n}_{2}}-1$ C) ${{n}_{1}}={{n}_{2}}$ D) ${{n}_{1}}>0,{{n}_{2}}>0$

Using ${{i}^{3}}=-i,{{i}^{5}}=i$ and${{i}^{7}}=-i$, we can write the given expression as ${{(1+i)}^{{{n}_{1}}}}+{{(1-i)}^{{{n}_{1}}}}+{{(1+i)}^{{{n}_{2}}}}+{{(1-i)}^{{{n}_{2}}}}$ $=2{{[}^{{{n}_{1}}}}{{C}_{0}}{{+}^{{{n}_{1}}}}{{C}_{2}}{{i}^{2}}{{+}^{{{n}_{1}}}}{{C}_{4}}{{i}^{4}}+.....]$$+2{{[}^{{{n}_{2}}}}{{C}_{0}}{{+}^{{{n}_{2}}}}{{C}_{2}}{{i}^{2}}{{+}^{{{n}_{2}}}}{{C}_{4}}{{i}^{4}}+.....]$   $=2{{[}^{{{n}_{1}}}}{{C}_{0}}{{-}^{{{n}_{1}}}}{{C}_{2}}{{+}^{{{n}_{1}}}}{{C}_{4}}+....]$ $+2{{[}^{{{n}_{2}}}}{{C}_{0}}{{-}^{{{n}_{2}}}}{{C}_{2}}{{+}^{{{n}_{2}}}}{{C}_{4}}+....]$ This is a real number irrespective of the values of ${{n}_{1}}$and ${{n}_{2}}$.