Question

For positive integers $n_1,n_2$ the value of the expression $(1+i{)}^{n_1}+(1+i^3{)}^{n_1}+(1+i^5{)}^{n_2}+(1+i^7{)}^{n_2},i=\sqrt{-1}$ 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 A $n_1=n_2+1$

$i^1=i$

$i^3=-i$

$i^5=i$

$i^7=-i$

$(1+i{)}^{n_1}+(1+i^3{)}^{n_1}+(1+i^5{)}^{n_2}+(1+i^7{)}^{n_2}=(1+i{)}^{n_1}+(1-i{)}^{n_1}+(1+i{)}^{n_2}+(1-i{)}^{n_2}$

As we know

$(1+i{)}^{n_1}=C_0+C_{1}i+C_2{i}^2+C_3{i}^3+......$ (a)

$(1-i{)}^{n_1}=C_0-C_{1}i+C_2{i}^2-C_3{i}^3+......$ (b)

$n_1$ is positive and constants are real

Adding (a) and (b)

$(1+i{)}^{n_1}+(1-i{)}^{n_1}=2(C_0+C_2{i}^2+C_4{i}^4.....)$

$(1+i{)}^{n_1}+(1-i{)}^{n_1}=2(C_0-C_2+C_4.....)=real$

Similarly for $n_2$

So

$(1+i{)}^{n_1}+(1+i^3{)}^{n_1}+(1+i^5{)}^{n_2}+(1+i^7{)}^{n_2}$ is real for all $n_1$ and $n_2$ provided they are positive.