Question

For positive integers 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

• A. $n_1=n_2+1$
• B. $n_1=n_2+1$
• C. $n_1=n_2$
• D. $n_1>0,n_2>0$

Answer 1

The correct option is D $n_1>0,n_2>0$

Using $i^3=-i,i^5=i$ and $i^7=-1$, we can write the given expression as $(1+i{)}^{n_1}+(1-i{)}^{n_1}+(1+i{)}^{n_2}+(1-i{)}^{n_2}$
$=2{[}^{n_1}{c}_0{+}^{n_1}{c}_4{i}^4+.....]+2{[}^{n_2}{c}_0{+}^{n_2}{C}_2{i}^2{+}^{n_2}{C}_4{i}^4.....]=2{[}^{n_2}{C}_0{-}^{n_1}{C}_2{+}^{n_1}{C}_4+....]+2{[}^{n_2}{C}_0{-}^{n_2}{C}_2{+}^{n_2}{C}_4+....]$
This is a real number irrespective of the values of $n_1$and $n_2$.