Question

For positive integers n₁,n₂ the value of the expression

is a real number if and only if ($1+i{)}^{n_1}$+($1+i^3{)}^{n_1}$+($1+i^5{)}^{n_2}$($1+i^7{)}^{n_2}$ where i=$\sqrt[2]{2-1}$ is a real number if and only if

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Using $i^3$=-i,$i^5$=i and $i^7$=-i, we can write the

given expression as

($1+i{)}^{n_1}$+($1+i^3{)}^{n_1}$+($1+i^5{)}^{n_2}$($1+i^7{)}^{n_2}$ where i=$\sqrt[2]{2-1}$

=2[$c_0^{n_1}$+$c_1^{n_1}$$i^2$+$c_1^{n_1}$$i^4$+....]+2[$c_0^{n_2}$+$c_1^{n_2}$$i^2$+$c_1^{n_2}$$i^4$+....]

=2[$c_0^{n_1}$-$c_1^{n_1}$+$c_1^{n_1}$+....]+2[$c_0^{n_2}$-$c_1^{n_2}$+$c_1^{n_2}$+....]

This is a real number irrespective of values of $n_1$ and $n_2$.