Complex Numbers And Quadratic Equations question 126
Question: 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]
Options:
A) $ n_1=n_2+1 $
B) $ n_1=n_2-1 $
C) $ n_1=n_2 $
D) $ n_1>0,n_2>0 $
Show Answer
Answer:
Correct Answer: D
Solution:
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_2i^{2}{{+}^{n_1}}C_4i^{4}+…..] $ $ +2{{[}^{n_2}}C_0{{+}^{n_2}}C_2i^{2}{{+}^{n_2}}C_4i^{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 $ .
BETA
